id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-23400 | Magnetic Field of Infinite Straight Conductor carrying Steady Current | Let $s$ be an infinitely long wire lying along the $z$ axis and carrying a steady current.
Let $\map {\mathbf B} {\mathbf r}$ be the magnetic field at a position $\mathbf r$ due to $s$.
Then:
:$\map {\mathbf B} {\mathbf r} = \dfrac {\mu_0 I} {2 \pi \rho} \paren {-\sin \theta \, \mathbf i + \cos \theta \, \mathbf j}$
wh... | We have a generic magnetic field:
:$\map {\mathbf B} {\mathbf r} = \begin{bmatrix} \map {B_x} {x, y, z} \\ \map {B_y} {x, y, z} \\ \map {B_z} {x, y, z} \\ \end{bmatrix}$
Due to translational symmetry in the $z$ direction, the magnetic field is independent of the $z$ coordinate.
:$\map {\mathbf B} {\mathbf r} = \begin{b... | Let $s$ be an [[Definition:Infinite|infinitely]] [[Definition:Length of Line|long]] [[Definition:Wire (Electricity)|wire]] lying along the [[Definition:Z-Axis|$z$ axis]] and carrying a [[Definition:Steady Current|steady current]].
Let $\map {\mathbf B} {\mathbf r}$ be the [[Definition:Magnetic Field|magnetic field]] a... | We have a generic [[Definition:Magnetic Field|magnetic field]]:
:$\map {\mathbf B} {\mathbf r} = \begin{bmatrix} \map {B_x} {x, y, z} \\ \map {B_y} {x, y, z} \\ \map {B_z} {x, y, z} \\ \end{bmatrix}$
Due to [[Definition:Translational Symmetry|translational symmetry]] in the $z$ [[Definition:Direction|direction]], the... | Magnetic Field of Infinite Straight Conductor carrying Steady Current/Proof 2 | https://proofwiki.org/wiki/Magnetic_Field_of_Infinite_Straight_Conductor_carrying_Steady_Current | https://proofwiki.org/wiki/Magnetic_Field_of_Infinite_Straight_Conductor_carrying_Steady_Current/Proof_2 | [
"Magnetic Field of Infinite Straight Conductor carrying Steady Current",
"Electric Current",
"Magnetic Fields",
"Electromagnetism"
] | [
"Definition:Infinite",
"Definition:Linear Measure/Length",
"Definition:Wire (Electricity)",
"Definition:Axis/Z-Axis",
"Definition:Steady Current",
"Definition:Magnetic Field",
"Definition:Position Vector",
"Definition:Vacuum Permeability",
"Definition:Electric Current",
"Definition:Perpendicular D... | [
"Definition:Magnetic Field",
"Definition:Translational Symmetry",
"Definition:Direction",
"Definition:Magnetic Field",
"Definition:Coordinate System/Coordinate",
"Definition:Rotational Symmetry",
"Definition:Axis/Z-Axis",
"Definition:Magnetic Field",
"Definition:Point",
"Definition:Rotation (Geome... |
proofwiki-23401 | Magnetic Force on Conductor carrying Steady Current | Let $s$ be a wire carrying a steady electric current $I$.
Let $\d \mathbf l$ be an infinitesimal length of the wire that points in the direction of the current.
Let $\mathbf B$ be the magnetic field acting at $\d \mathbf l$.
Then, the force $\d \mathbf F$ acting on $\d \mathbf l$ is
:$\d \mathbf F = I \d \mathbf l \tim... | According to Magnetic Force on Moving Charged Particle, a single particle with charge $q$ moving with velocity $\mathbf v$ in a magnetic field $\mathbf B$ experiences a force of:
:$\mathbf F = q \mathbf v \times \mathbf B$
For a current in a wire, we will consider an infinitesimal amount of charge $\d q$ feeling an in... | Let $s$ be a [[Definition:Wire (Electricity)|wire]] carrying a [[Definition:Steady Current|steady electric current]] $I$.
Let $\d \mathbf l$ be an [[Definition:Infinitesimal|infinitesimal]] [[Definition:Length of Line|length]] of the [[Definition:Wire (Electricity)|wire]] that points in the [[Definition:Direction|dire... | According to [[Magnetic Force on Moving Charged Particle]], a single [[Definition:Particle|particle]] with [[Definition:Electric Charge|charge]] $q$ [[Definition:Motion|moving]] with [[Definition:Velocity|velocity]] $\mathbf v$ in a [[Definition:Magnetic Field|magnetic field]] $\mathbf B$ experiences a [[Definition:Fo... | Magnetic Force on Conductor carrying Steady Current | https://proofwiki.org/wiki/Magnetic_Force_on_Conductor_carrying_Steady_Current | https://proofwiki.org/wiki/Magnetic_Force_on_Conductor_carrying_Steady_Current | [
"Electromagnetism",
"Magnetic Forces",
"Electric Current"
] | [
"Definition:Wire (Electricity)",
"Definition:Steady Current",
"Definition:Infinitesimal",
"Definition:Linear Measure/Length",
"Definition:Wire (Electricity)",
"Definition:Direction",
"Definition:Electric Current",
"Definition:Magnetic Field",
"Definition:Force",
"Definition:Vector Cross Product"
] | [
"Magnetic Force on Moving Charged Particle",
"Definition:Particle",
"Definition:Electric Charge",
"Definition:Motion",
"Definition:Velocity",
"Definition:Magnetic Field",
"Definition:Force",
"Definition:Electric Current",
"Definition:Wire (Electricity)",
"Definition:Infinitesimal",
"Definition:E... |
proofwiki-23402 | Topology Induced by Metric is Unique | Let $M = \struct {A, d}$ be a metric space.
Let $T = \struct {A, \tau_d}$ be the topological space such that $\tau_d$ is the topology induced by $d$.
Then $\tau_d$ is unique.
That is, if $\tau_{d 1}$ and $\tau_{d 2}$ are both induced by $d$, then $\tau_{d 1} = \tau_{d 2}$. | {{Recall|Topology Induced by Metric|topology induced by $d$}}
{{:Definition:Topology Induced by Metric/Definition 1}}
There is only one set of open sets of $M$.
Hence by definition there is only one set of open sets of $T$.
But by definition of topology, $\tau_d$ ''is'' the set of open sets of $T$.
The result follows.
... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $T = \struct {A, \tau_d}$ be the [[Definition:Topological Space|topological space]] such that $\tau_d$ is the [[Definition:Topology Induced by Metric|topology induced by $d$]].
Then $\tau_d$ is [[Definition:Unique|unique]].
That is, if $\ta... | {{Recall|Topology Induced by Metric|topology induced by $d$}}
{{:Definition:Topology Induced by Metric/Definition 1}}
There is only one [[Definition:Set|set]] of [[Definition:Open Set of Metric Space|open sets]] of $M$.
Hence by definition there is only one [[Definition:Set|set]] of [[Definition:Open Set (Topology)|o... | Topology Induced by Metric is Unique | https://proofwiki.org/wiki/Topology_Induced_by_Metric_is_Unique | https://proofwiki.org/wiki/Topology_Induced_by_Metric_is_Unique | [
"Topologies Induced by Metrics"
] | [
"Definition:Metric Space",
"Definition:Topological Space",
"Definition:Topology Induced by Metric",
"Definition:Unique",
"Definition:Topology Induced by Metric"
] | [
"Definition:Set",
"Definition:Open Set/Metric Space",
"Definition:Set",
"Definition:Open Set/Topology",
"Definition:Topology",
"Definition:Set",
"Definition:Open Set/Topology"
] |
proofwiki-23403 | Metric inducing Topology is not Unique | Let $T = \struct {A, \tau}$ be a topological space such that its topology $\tau$ is induced by a metric on $A$.
Then that metric is specifically ''not'' unique.
In other words, given that $\tau$ is the induced by a metric $d_1$ on $A$, there exists another metric $d_2$ on $A$ such that $d_2$ also induces topology $\tau... | A direct consequence of:
:Construction of Infinite Set of Metrics yielding Same Metric Space
:Metric Induces Topology
{{qed}} | Let $T = \struct {A, \tau}$ be a [[Definition:Topological Space|topological space]] such that its [[Definition:Topology|topology]] $\tau$ is [[Definition:Topology Induced by Metric|induced]] by a [[Definition:Metric|metric]] on $A$.
Then that [[Definition:Metric|metric]] is specifically ''not'' [[Definition:Unique|uni... | A direct consequence of:
:[[Construction of Infinite Set of Metrics yielding Same Metric Space]]
:[[Metric Induces Topology]]
{{qed}} | Metric inducing Topology is not Unique | https://proofwiki.org/wiki/Metric_inducing_Topology_is_not_Unique | https://proofwiki.org/wiki/Metric_inducing_Topology_is_not_Unique | [
"Topologies Induced by Metrics"
] | [
"Definition:Topological Space",
"Definition:Topology",
"Definition:Topology Induced by Metric",
"Definition:Metric Space/Metric",
"Definition:Metric Space/Metric",
"Definition:Unique",
"Definition:Topology Induced by Metric",
"Definition:Metric Space/Metric",
"Definition:Metric Space/Metric",
"Def... | [
"Construction of Infinite Set of Metrics yielding Same Metric Space",
"Metric Induces Topology"
] |
proofwiki-23404 | Construction of Infinite Set of Metrics yielding Same Metric Space | Let $M = \struct {A, d}$ be a metric space.
Let $\SS$ denote the set of open sets of $M$.
Then there exists an infinite set of metrics on $A$ such that the set of open sets of the resulting metric space is also $\SS$.
That is, given a metric space, there is an infinite number of metrics which yield that same metric spa... | ;Outline of Proof
Let $M = \struct {A, d}$ be a metric space.
Let $\delta: A \to A$ be defined as:
:$\forall \tuple {x, y} \in A^2: \map \delta {x, y} := \dfrac {\map d {x, y} } {1 + \map d {x, y} }$
It is to be shown that $\delta$ is a metric on $A$.
Then it is to be shown that the open sets of $\struct {A, \delta}$ a... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\SS$ denote the [[Definition:Set|set]] of [[Definition:Open Set (Metric Space)|open sets]] of $M$.
Then there exists an [[Definition:Infinite Set|infinite set]] of [[Definition:Metric|metrics]] on $A$ such that the [[Definition:Set|set]] of ... | ;Outline of Proof
Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\delta: A \to A$ be defined as:
:$\forall \tuple {x, y} \in A^2: \map \delta {x, y} := \dfrac {\map d {x, y} } {1 + \map d {x, y} }$
It is to be shown that $\delta$ is a [[Definition:Metric|metric]] on $A$.
Then it is to b... | Construction of Infinite Set of Metrics yielding Same Metric Space | https://proofwiki.org/wiki/Construction_of_Infinite_Set_of_Metrics_yielding_Same_Metric_Space | https://proofwiki.org/wiki/Construction_of_Infinite_Set_of_Metrics_yielding_Same_Metric_Space | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Set",
"Definition:Open Set/Metric Space",
"Definition:Infinite Set",
"Definition:Metric Space/Metric",
"Definition:Set",
"Definition:Open Set/Metric Space",
"Definition:Metric Space",
"Definition:Metric Space",
"Definition:Metric Space/Metric",
"Definition:... | [
"Definition:Metric Space",
"Definition:Metric Space/Metric",
"Definition:Open Set/Metric Space",
"Definition:Set",
"Definition:Open Set/Metric Space",
"Definition:Metric Space/Metric",
"Definition:Metric Space/Metric",
"Definition:Metric Space/Metric",
"Definition:Metric Space",
"Definition:Metric... |
proofwiki-23405 | Countably Compact Metric Space is Separable | Let $M$ be a countably compact metric space.
Then $M$ is separable. | This follows directly from the results:
:Countably Compact Metric Space is Sequentially Compact
:Sequentially Compact Metric Space is Separable.
{{qed}} | Let $M$ be a [[Definition:Countably Compact Space|countably compact]] [[Definition:Metric Space|metric space]].
Then $M$ is [[Definition:Separable Space|separable]]. | This follows directly from the results:
:[[Countably Compact Metric Space is Sequentially Compact]]
:[[Sequentially Compact Metric Space is Separable]].
{{qed}} | Countably Compact Metric Space is Separable | https://proofwiki.org/wiki/Countably_Compact_Metric_Space_is_Separable | https://proofwiki.org/wiki/Countably_Compact_Metric_Space_is_Separable | [
"Metric Spaces",
"Countably Compact Spaces",
"Separable Spaces"
] | [
"Definition:Countably Compact Space",
"Definition:Metric Space",
"Definition:Separable Space"
] | [
"Countably Compact Metric Space is Sequentially Compact",
"Sequentially Compact Metric Space is Separable"
] |
proofwiki-23406 | Countably Compact Metric Space is Second-Countable | Let $M$ be a countably compact metric space.
Then $M$ is second-countable. | This follows directly from the results:
:Countably Compact Metric Space is Separable
:Separable Metric Space is Second-Countable.
{{qed}} | Let $M$ be a [[Definition:Countably Compact Space|countably compact]] [[Definition:Metric Space|metric space]].
Then $M$ is [[Definition:Second-Countable Space|second-countable]]. | This follows directly from the results:
:[[Countably Compact Metric Space is Separable]]
:[[Separable Metric Space is Second-Countable]].
{{qed}} | Countably Compact Metric Space is Second-Countable | https://proofwiki.org/wiki/Countably_Compact_Metric_Space_is_Second-Countable | https://proofwiki.org/wiki/Countably_Compact_Metric_Space_is_Second-Countable | [
"Metric Spaces",
"Countably Compact Spaces",
"Second-Countable Spaces"
] | [
"Definition:Countably Compact Space",
"Definition:Metric Space",
"Definition:Second-Countable Space"
] | [
"Countably Compact Metric Space is Separable",
"Separable Metric Space is Second-Countable"
] |
proofwiki-23407 | Countably Compact Second-Countable Space is Compact | Let $T = \struct {S, \tau}$ be a countably compact second-countable space.
Then $T$ is compact. | {{Recall|Compact Topological Space|compact topological space}}
{{:Definition:Compact Topological Space/Definition 1}}
Let $T$ be countably compact and second-countable.
{{Recall|Countably Compact Space|countably compact space}}
{{:Definition:Countably Compact Space/Definition 1}}
{{Recall|Second-Countable Space|second-... | Let $T = \struct {S, \tau}$ be a [[Definition:Countably Compact Space|countably compact]] [[Definition:Second-Countable Space|second-countable space]].
Then $T$ is [[Definition:Compact Topological Space|compact]]. | {{Recall|Compact Topological Space|compact topological space}}
{{:Definition:Compact Topological Space/Definition 1}}
Let $T$ be [[Definition:Countably Compact Space|countably compact]] and [[Definition:Second-Countable Space|second-countable]].
{{Recall|Countably Compact Space|countably compact space}}
{{:Definition... | Countably Compact Second-Countable Space is Compact | https://proofwiki.org/wiki/Countably_Compact_Second-Countable_Space_is_Compact | https://proofwiki.org/wiki/Countably_Compact_Second-Countable_Space_is_Compact | [
"Countably Compact Spaces",
"Second-Countable Spaces",
"Compact Topological Spaces"
] | [
"Definition:Countably Compact Space",
"Definition:Second-Countable Space",
"Definition:Compact Topological Space"
] | [
"Definition:Countably Compact Space",
"Definition:Second-Countable Space",
"Definition:Second-Countable Space",
"Definition:Countable Basis",
"Definition:Open Cover",
"Definition:Countable Set",
"Definition:Basis (Topology)/Analytic Basis",
"Subset Relation is Transitive",
"Equivalence of Definition... |
proofwiki-23408 | Solenoidal Vector Field is Curl | Let $\mathbf V$ be a solenoidal vector field on a simply connected region of $\mathbb R^3$.
Then there exists a vector field $\mathbf A$ such that:
:$\mathbf V = \curl \mathbf A$ | {{Recall|Solenoidal Vector Field|solenoidal vector field|index = 1}}
{{:Definition:Solenoidal Vector Field/Definition 1}}
We seek any vector field $\mathbf A$ such that:
{{begin-eqn}}
{{eqn | l = \mathbf V
| r = \nabla \times \mathbf A
}}
{{eqn | r = \paren {\frac {\partial A_z} {\partial y} - \frac {\partial A_y... | Let $\mathbf V$ be a [[Definition:Solenoidal Vector Field|solenoidal vector field]] on a [[Definition:Simply Connected|simply connected]] [[Definition:Region|region]] of $\mathbb R^3$.
Then there exists a [[Definition:Vector Field|vector field]] $\mathbf A$ such that:
:$\mathbf V = \curl \mathbf A$ | {{Recall|Solenoidal Vector Field|solenoidal vector field|index = 1}}
{{:Definition:Solenoidal Vector Field/Definition 1}}
We seek any [[Definition:Vector Field|vector field]] $\mathbf A$ such that:
{{begin-eqn}}
{{eqn | l = \mathbf V
| r = \nabla \times \mathbf A
}}
{{eqn | r = \paren {\frac {\partial A_z} {\par... | Solenoidal Vector Field is Curl | https://proofwiki.org/wiki/Solenoidal_Vector_Field_is_Curl | https://proofwiki.org/wiki/Solenoidal_Vector_Field_is_Curl | [
"Solenoidal Vector Fields",
"Curl Operator",
"Divergence Operator",
"Gradient Operator"
] | [
"Definition:Solenoidal Vector Field",
"Definition:Simply Connected",
"Definition:Region",
"Definition:Vector Field"
] | [
"Definition:Vector Field",
"Definition:Real-Valued Function",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-23409 | Vector Fields with Equal Curls Differ by Gradient | Let $\mathbf A$ and $\mathbf B$ be vector fields over ordinary space.
Then:
:$\nabla \times \mathbf A = \nabla \times \mathbf B$
{{iff}}
:$\mathbf B = \mathbf A + \grad f$
where $f$ is a scalar function with continuous partial derivatives. | === Sufficient Condition ===
We have:
{{begin-eqn}}
{{eqn | l = \nabla \times \mathbf B
| r = \nabla \times \mathbf A
}}
{{eqn | ll= \leadsto
| l = \nabla \times \mathbf B - \nabla \times \mathbf A
| r = \mathbf 0
}}
{{eqn | ll= \leadsto
| l = \nabla \times \paren {\mathbf B - \mathbf A}
|... | Let $\mathbf A$ and $\mathbf B$ be [[Definition:Vector Field|vector fields]] over [[Definition:Ordinary Space|ordinary space]].
Then:
:$\nabla \times \mathbf A = \nabla \times \mathbf B$
{{iff}}
:$\mathbf B = \mathbf A + \grad f$
where $f$ is a [[Definition:Real-Valued Function|scalar function]] with [[Definition:Cont... | === Sufficient Condition ===
We have:
{{begin-eqn}}
{{eqn | l = \nabla \times \mathbf B
| r = \nabla \times \mathbf A
}}
{{eqn | ll= \leadsto
| l = \nabla \times \mathbf B - \nabla \times \mathbf A
| r = \mathbf 0
}}
{{eqn | ll= \leadsto
| l = \nabla \times \paren {\mathbf B - \mathbf A}
... | Vector Fields with Equal Curls Differ by Gradient | https://proofwiki.org/wiki/Vector_Fields_with_Equal_Curls_Differ_by_Gradient | https://proofwiki.org/wiki/Vector_Fields_with_Equal_Curls_Differ_by_Gradient | [
"Curl Operator",
"Gradient Operator"
] | [
"Definition:Vector Field",
"Definition:Ordinary Space",
"Definition:Real-Valued Function",
"Definition:Continuous",
"Definition:Partial Derivative"
] | [
"Vector Cross Product Operator is Bilinear",
"Definition:Conservative Vector Field",
"Definition:Existential Quantifier",
"Definition:Real-Valued Function"
] |
proofwiki-23410 | Open Unit Interval on Rational Number Space is not Compact | Let $T = \struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then:
:$\openint 0 1 \cap \Q$ is not compact in $T$
where $\openint 0 1$ is the open unit interval. | From the Heine-Borel Theorem on a metric space, $\openint 0 1 \cap \Q$ is compact {{iff}} it is both totally bounded and complete.
From Rational Number Space is not Complete Metric Space it follows that $\openint 0 1 \cap \Q$ is not compact.
{{qed}} | Let $T = \struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then:
:$\openint 0 1 \cap \Q$ is not [[Definition:Compact Metric Space|compact]] in $T$
where $\openint 0 1$ is the [[Definition:... | From the [[Heine-Borel Theorem/Metric Space|Heine-Borel Theorem on a metric space]], $\openint 0 1 \cap \Q$ is [[Definition:Compact Metric Space|compact]] {{iff}} it is both [[Definition:Totally Bounded Metric Space|totally bounded]] and [[Definition:Complete Metric Space|complete]].
From [[Rational Number Space is no... | Open Unit Interval on Rational Number Space is not Compact | https://proofwiki.org/wiki/Open_Unit_Interval_on_Rational_Number_Space_is_not_Compact | https://proofwiki.org/wiki/Open_Unit_Interval_on_Rational_Number_Space_is_not_Compact | [
"Rational Number Space",
"Examples of Compact Metric Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Compact Space/Metric Space",
"Definition:Real Interval/Unit Interval/Open"
] | [
"Heine-Borel Theorem/Metric Space",
"Definition:Compact Space/Metric Space",
"Definition:Totally Bounded Metric Space",
"Definition:Complete Metric Space",
"Rational Number Space is not Complete Metric Space",
"Definition:Compact Space/Metric Space"
] |
proofwiki-23411 | Totally Bounded Metric Space is not necessarily Compact | A metric space which is totally bounded is not necessarily also compact. | Let $T = \struct {\closedint 0 1 \cap \Q, \tau_d}$ be the closed unit interval of rational numbers under the usual (Euclidean) topology.
From Open Unit Interval on Rational Number Space is Totally Bounded, $T$ is a totally bounded space.
But from Open Unit Interval on Rational Number Space is not Compact, $T$ is not a ... | A [[Definition:Metric Space|metric space]] which is [[Definition:Totally Bounded Metric Space|totally bounded]] is not necessarily also [[Definition:Compact Metric Space|compact]]. | Let $T = \struct {\closedint 0 1 \cap \Q, \tau_d}$ be the [[Definition:Closed Unit Interval|closed unit interval]] of [[Definition:Rational Number|rational numbers]] under the [[Definition:Usual Topology|usual (Euclidean) topology]].
From [[Open Unit Interval on Rational Number Space is Totally Bounded]], $T$ is a [[... | Totally Bounded Metric Space is not necessarily Compact | https://proofwiki.org/wiki/Totally_Bounded_Metric_Space_is_not_necessarily_Compact | https://proofwiki.org/wiki/Totally_Bounded_Metric_Space_is_not_necessarily_Compact | [
"Totally Bounded Metric Spaces",
"Compact Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Totally Bounded Metric Space",
"Definition:Compact Space/Metric Space"
] | [
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Rational Number",
"Definition:Euclidean Space/Euclidean Topology",
"Open Unit Interval on Rational Number Space is Totally Bounded",
"Definition:Totally Bounded Metric Space",
"Open Unit Interval on Rational Number Space is not Compact",
"Defi... |
proofwiki-23412 | Second-Countable Metric Space is not necessarily Totally Bounded | A metric space which is second-countable is not necessarily also totally bounded. | {{ProofWanted|somewhere down in the depths of item $134$}} | A [[Definition:Metric Space|metric space]] which is [[Definition:Second-Countable Space|second-countable]] is not necessarily also [[Definition:Totally Bounded Metric Space|totally bounded]]. | {{ProofWanted|somewhere down in the depths of item $134$}} | Second-Countable Metric Space is not necessarily Totally Bounded | https://proofwiki.org/wiki/Second-Countable_Metric_Space_is_not_necessarily_Totally_Bounded | https://proofwiki.org/wiki/Second-Countable_Metric_Space_is_not_necessarily_Totally_Bounded | [
"Totally Bounded Metric Spaces",
"Second-Countable Spaces",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Second-Countable Space",
"Definition:Totally Bounded Metric Space"
] | [] |
proofwiki-23413 | Complete Metric Space is not necessarily Compact | Let $M = \struct {A, d}$ be a complete metric space.
Then it is not necessarily the case that $M$ is also compact. | Let $M = \struct {\R, d}$ denote the real number line under the Euclidean metric.
From Real Number Line is Complete Metric Space, $M$ is a complete metric space.
From Real Number Line is not Compact, $M$ is not compact.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Complete Metric Space|complete metric space]].
Then it is not necessarily the case that $M$ is also [[Definition:Compact Metric Space|compact]]. | Let $M = \struct {\R, d}$ denote the [[Definition:Real Number Line|real number line]] under the [[Definition:Euclidean Metric|Euclidean metric]].
From [[Real Number Line is Complete Metric Space]], $M$ is a [[Definition:Complete Metric Space|complete metric space]].
From [[Real Number Line is not Compact]], $M$ is no... | Complete Metric Space is not necessarily Compact | https://proofwiki.org/wiki/Complete_Metric_Space_is_not_necessarily_Compact | https://proofwiki.org/wiki/Complete_Metric_Space_is_not_necessarily_Compact | [
"Complete Metric Spaces",
"Compact Metric Spaces"
] | [
"Definition:Complete Metric Space",
"Definition:Compact Space/Metric Space"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Metric",
"Real Number Line is Complete Metric Space",
"Definition:Complete Metric Space",
"Real Number Line is not Compact",
"Definition:Compact Space/Metric Space"
] |
proofwiki-23414 | Completely Metrizable Space is Non-Meager | Let $T = \struct {S, \tau}$ be a completely metrizable topological space.
Then $T$ is non-meager. | Let $T = \struct {S, \tau}$ be completely metrizable.
By Complete Metric Space is Non-Meager, it follows that $T$ is non-meager.
{{qed}}
{{ADC|Complete Metric Space is Non-Meager}} | Let $T = \struct {S, \tau}$ be a [[Definition:Completely Metrizable Space|completely metrizable topological space]].
Then $T$ is [[Definition:Non-Meager Space|non-meager]]. | Let $T = \struct {S, \tau}$ be [[Definition:Completely Metrizable Space|completely metrizable]].
By [[Complete Metric Space is Non-Meager]], it follows that $T$ is [[Definition:Non-Meager Space|non-meager]].
{{qed}}
{{ADC|Complete Metric Space is Non-Meager}} | Completely Metrizable Space is Non-Meager | https://proofwiki.org/wiki/Completely_Metrizable_Space_is_Non-Meager | https://proofwiki.org/wiki/Completely_Metrizable_Space_is_Non-Meager | [
"Completely Metrizable Spaces",
"Non-Meager Spaces",
"Examples of Use of Axiom of Dependent Choice"
] | [
"Definition:Completely Metrizable Space",
"Definition:Meager Space/Non-Meager"
] | [
"Definition:Completely Metrizable Space",
"Complete Metric Space is Non-Meager",
"Definition:Meager Space/Non-Meager"
] |
proofwiki-23415 | Regular Lindelöf Space is Paracompact | Let $T = \struct {S, \tau}$ be a regular space which is also Lindelöf.
Then $T$ is paracompact. | Let $T = \struct {S, \tau}$ be a regular Lindelöf space.
{{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
Hence $T$ is {{afortiori}} a $T_3$ space.
The result follows from $T_3$ Lindelöf Space is Paracompact.
{{qed}}
Category:Regular Spaces
Category:Lindelöf Spaces
Category:Paracompact Spac... | Let $T = \struct {S, \tau}$ be a [[Definition:Regular Space|regular space]] which is also [[Definition:Lindelöf Space|Lindelöf]].
Then $T$ is [[Definition:Paracompact Space|paracompact]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Regular Space|regular]] [[Definition:Lindelöf Space|Lindelöf space]].
{{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
Hence $T$ is {{afortiori}} a [[Definition:T3 Space|$T_3$ space]].
The result follows from [[T3 Lindelöf Space is Paracompac... | Regular Lindelöf Space is Paracompact | https://proofwiki.org/wiki/Regular_Lindelöf_Space_is_Paracompact | https://proofwiki.org/wiki/Regular_Lindelöf_Space_is_Paracompact | [
"Regular Spaces",
"Lindelöf Spaces",
"Paracompact Spaces"
] | [
"Definition:Regular Space",
"Definition:Lindelöf Space",
"Definition:Paracompact Space"
] | [
"Definition:Regular Space",
"Definition:Lindelöf Space",
"Definition:T3 Space",
"T3 Lindelöf Space is Paracompact",
"Category:Regular Spaces",
"Category:Lindelöf Spaces",
"Category:Paracompact Spaces"
] |
proofwiki-23416 | Radial Interval Space is Completely Normal | The radial interval space is completely normal. | {{ProofWanted}}
Category:Radial Interval Topology
Category:Examples of Completely Normal Spaces
4yznhgz97rip3vji7iskx6vpvdb58sb | The [[Definition:Radial Interval Space|radial interval space]] is [[Definition:Completely Normal Space|completely normal]]. | {{ProofWanted}}
[[Category:Radial Interval Topology]]
[[Category:Examples of Completely Normal Spaces]]
4yznhgz97rip3vji7iskx6vpvdb58sb | Radial Interval Space is Completely Normal | https://proofwiki.org/wiki/Radial_Interval_Space_is_Completely_Normal | https://proofwiki.org/wiki/Radial_Interval_Space_is_Completely_Normal | [
"Radial Interval Topology",
"Examples of Completely Normal Spaces",
"Radial Interval Topology",
"Examples of Completely Normal Spaces"
] | [
"Definition:Radial Interval Topology",
"Definition:Completely Normal Space"
] | [
"Category:Radial Interval Topology",
"Category:Examples of Completely Normal Spaces"
] |
proofwiki-23417 | Radial Interval Space is Paracompact | The radial interval space is paracompact. | {{ProofWanted}}
Category:Radial Interval Topology
Category:Examples of Paracompact Spaces
anc1qwefhyprr135ajotf9zxd565e5j | The [[Definition:Radial Interval Space|radial interval space]] is [[Definition:Paracompact Space|paracompact]]. | {{ProofWanted}}
[[Category:Radial Interval Topology]]
[[Category:Examples of Paracompact Spaces]]
anc1qwefhyprr135ajotf9zxd565e5j | Radial Interval Space is Paracompact | https://proofwiki.org/wiki/Radial_Interval_Space_is_Paracompact | https://proofwiki.org/wiki/Radial_Interval_Space_is_Paracompact | [
"Radial Interval Topology",
"Examples of Paracompact Spaces",
"Radial Interval Topology",
"Examples of Paracompact Spaces"
] | [
"Definition:Radial Interval Topology",
"Definition:Paracompact Space"
] | [
"Category:Radial Interval Topology",
"Category:Examples of Paracompact Spaces"
] |
proofwiki-23418 | Radial Interval Space is not Metrizable | The radial interval space is not metrizable. | {{ProofWanted}}
Category:Radial Interval Topology
Category:Examples of Metrizable Spaces
sq66j2lag692btpvhf6g1obdljw2m57 | The [[Definition:Radial Interval Space|radial interval space]] is not [[Definition:Metrizable Space|metrizable]]. | {{ProofWanted}}
[[Category:Radial Interval Topology]]
[[Category:Examples of Metrizable Spaces]]
sq66j2lag692btpvhf6g1obdljw2m57 | Radial Interval Space is not Metrizable | https://proofwiki.org/wiki/Radial_Interval_Space_is_not_Metrizable | https://proofwiki.org/wiki/Radial_Interval_Space_is_not_Metrizable | [
"Radial Interval Topology",
"Examples of Metrizable Spaces"
] | [
"Definition:Radial Interval Topology",
"Definition:Metrizable Space"
] | [
"Category:Radial Interval Topology",
"Category:Examples of Metrizable Spaces"
] |
proofwiki-23419 | Quasiuniformity Inducing Topology may not be Unique | Let $T = \struct {S, \tau}$ be a topological space.
Let $\UU$ be a quasiuniformity on a set $S$ such that $T$ is the topology induced by $\UU$.
Then it is not necessarily the case that $\UU$ is the only quasiuniformity that induces $T$. | {{ProofWanted|we look at item $44$ in S&S}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\UU$ be a [[Definition:Quasiuniformity|quasiuniformity]] on a [[Definition:Set|set]] $S$ such that $T$ is the [[Definition:Topology Induced by Quasiuniformity|topology induced]] by $\UU$.
Then it is not necessarily the case tha... | {{ProofWanted|we look at item $44$ in S&S}} | Quasiuniformity Inducing Topology may not be Unique | https://proofwiki.org/wiki/Quasiuniformity_Inducing_Topology_may_not_be_Unique | https://proofwiki.org/wiki/Quasiuniformity_Inducing_Topology_may_not_be_Unique | [
"Topologies Induced by Quasiuniformities",
"Quasiuniformities"
] | [
"Definition:Topological Space",
"Definition:Quasiuniformity",
"Definition:Set",
"Definition:Topology Induced by Quasiuniformity",
"Definition:Quasiuniformity",
"Definition:Topology Induced by Quasiuniformity"
] | [] |
proofwiki-23420 | Spectral Theorem for Hermitian Matrices | Let $\mathbf A$ be a square matrix over $\mathbb C$.
Then $\mathbf A$ is Hermitian {{Iff}} it is diagonalizable to a real diagonal matrix via a unitary transformation.
That is, we can write:
:$\mathbf A = \mathbf U \mathbf D \mathbf U^\dagger$
where:
:$\mathbf D$ is a real diagonal matrix
:$\mathbf U$ is a unitary matr... | === Necessary Case ===
Let $\mathbf A$ be diagonalizable to a real diagonal matrix via a unitary transformation.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A^\dagger
| r = \paren {\mathbf U \mathbf D \mathbf U^\dagger}^\dagger
}}
{{eqn | r = \mathbf U \mathbf D^\dagger \mathbf U^\dagger
| c = Hermitian Conjuga... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] over $\mathbb C$.
Then $\mathbf A$ is [[Definition:Hermitian Matrix|Hermitian]] {{Iff}} it is [[Definition:Diagonalizable Matrix|diagonalizable]] to a [[Definition:Real Matrix|real]] [[Definition:Diagonal Matrix|diagonal matrix]] via a [[Definition:Unitar... | === Necessary Case ===
Let $\mathbf A$ be [[Definition:Diagonalizable Matrix|diagonalizable]] to a [[Definition:Real Matrix|real]] [[Definition:Diagonal Matrix|diagonal matrix]] via a [[Definition:Unitary Transformation|unitary transformation]].
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A^\dagger
| r = \paren {\m... | Spectral Theorem for Hermitian Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Hermitian_Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Hermitian_Matrices | [
"Spectral Theorems",
"Hermitian Matrices",
"Diagonalizable Matrices",
"Unitary Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Hermitian Matrix",
"Definition:Diagonalizable Matrix",
"Definition:Real Matrix",
"Definition:Diagonal Matrix",
"Definition:Unitary Transformation",
"Definition:Real Matrix",
"Definition:Diagonal Matrix",
"Definition:Unitary Matrix",
"Definition:Hermit... | [
"Definition:Diagonalizable Matrix",
"Definition:Real Matrix",
"Definition:Diagonal Matrix",
"Definition:Unitary Transformation",
"Hermitian Conjugate of Matrix Product",
"Hermitian Conjugate is Involution",
"Definition:Real Matrix",
"Definition:Diagonal Matrix",
"Definition:Hermitian Matrix",
"Def... |
proofwiki-23421 | Set of Real Orthogonal Matrices is Compact in Real Square Matrix Vector Space | let $n$ be a positive integer.
Then, $\mathrm O (n, \mathbb R)$ is a compact subset of the vector space $\mathcal M _n(\mathbb R)$ | {{Proofread}} | let $n$ be a positive integer.
Then, $\mathrm O (n, \mathbb R)$ is a [[Definition:Compact Space/Normed Vector Space|compact subset]] of the vector space $\mathcal M _n(\mathbb R)$ | {{Proofread}} | Set of Real Orthogonal Matrices is Compact in Real Square Matrix Vector Space | https://proofwiki.org/wiki/Set_of_Real_Orthogonal_Matrices_is_Compact_in_Real_Square_Matrix_Vector_Space | https://proofwiki.org/wiki/Set_of_Real_Orthogonal_Matrices_is_Compact_in_Real_Square_Matrix_Vector_Space | [
"Compact Normed Vector Spaces"
] | [
"Definition:Compact Space/Normed Vector Space"
] | [] |
proofwiki-23422 | Definition:QR Factorization:Existance Of QR Factorization For Square Matrices | Let $\mathbf A$ be a real square matrix.
Then $\mathbf A$ has a QR factorization, that is we can write
: $\mathbf A = \mathbf Q \mathbf R $
where
: $\mathbf Q$ is orthogonal
: $\mathbf R$ is upper triangular | {{ProofWanted}}
jrdtx2oedodevj8bfmqgp7tk3nlkfhu | Let $\mathbf A$ be a [[Definition:Real Matrix|real]] [[Definition:Matrix/Square Matrix|square matrix]].
Then $\mathbf A$ has a [[Definition:QR Factorization|QR factorization]], that is we can write
: $\mathbf A = \mathbf Q \mathbf R $
where
: $\mathbf Q$ is [[Definition:Orthogonal Matrix|orthogonal]]
: $\mathbf R$ is... | {{ProofWanted}}
jrdtx2oedodevj8bfmqgp7tk3nlkfhu | Definition:QR Factorization:Existance Of QR Factorization For Square Matrices | https://proofwiki.org/wiki/Definition:QR_Factorization:Existance_Of_QR_Factorization_For_Square_Matrices | https://proofwiki.org/wiki/Definition:QR_Factorization:Existance_Of_QR_Factorization_For_Square_Matrices | [] | [
"Definition:Real Matrix",
"Definition:Matrix/Square Matrix",
"Definition:QR Factorization",
"Definition:Orthogonal Matrix",
"Definition:Triangular Matrix/Upper Triangular Matrix"
] | [] |
proofwiki-23423 | Product Formula for Cosine | :$\ds \map \cos {n z} = \paren {-1}^{\floor {n/2} } 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \cos {z + \frac {\paren {\map {\cos^2} {\frac {n \pi} 2} + 2 k} \pi} {2 n} }$ | === Lemma 1 ===
{{:Product Formula for Cosine/Lemma 1}}{{qed|lemma}} | :$\ds \map \cos {n z} = \paren {-1}^{\floor {n/2} } 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \cos {z + \frac {\paren {\map {\cos^2} {\frac {n \pi} 2} + 2 k} \pi} {2 n} }$ | === [[Product Formula for Cosine/Lemma 1|Lemma 1]] ===
{{:Product Formula for Cosine/Lemma 1}}{{qed|lemma}} | Product Formula for Cosine | https://proofwiki.org/wiki/Product_Formula_for_Cosine | https://proofwiki.org/wiki/Product_Formula_for_Cosine | [
"Product Formula for Cosine",
"Cosine Function"
] | [] | [
"Product Formula for Cosine/Lemma 1"
] |
proofwiki-23424 | Trivial Topological Space has no Limit Point | Let $T = \struct {S, \tau}$ be a trivial topological space.
Then $T$ has no limit points. | By definition of trivial topological space, $S$ is a singleton: $S = \set s$, say.
{{Recall|Limit Point of Set/Definition from Open Neighborhood|limit point of $H \subseteq S$}}
{{:Definition:Limit Point of Set/Definition from Open Neighborhood}}
As $S$ is a singleton, the only non-empty subset of $S$ is $S$ itself.
As... | Let $T = \struct {S, \tau}$ be a [[Definition:Trivial Topological Space|trivial topological space]].
Then $T$ has no [[Definition:Limit Point of Set|limit points]]. | By definition of [[Definition:Trivial Topological Space|trivial topological space]], $S$ is a [[Definition:Singleton|singleton]]: $S = \set s$, say.
{{Recall|Limit Point of Set/Definition from Open Neighborhood|limit point of $H \subseteq S$}}
{{:Definition:Limit Point of Set/Definition from Open Neighborhood}}
As $S... | Trivial Topological Space has no Limit Point | https://proofwiki.org/wiki/Trivial_Topological_Space_has_no_Limit_Point | https://proofwiki.org/wiki/Trivial_Topological_Space_has_no_Limit_Point | [
"Trivial Topological Spaces",
"Examples of Limit Points"
] | [
"Definition:Trivial Topological Space",
"Definition:Limit Point/Topology/Set"
] | [
"Definition:Trivial Topological Space",
"Definition:Singleton",
"Definition:Singleton",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Element",
"Definition:Element",
"Definition:Distinct/Plural",
"Category:Trivial Topological Spaces",
"Category:Examples of Limit Points"
] |
proofwiki-23425 | Finite Particular Point Space is Weakly Countably Compact | Let $T = \struct {S, \tau_p}$ be a finite particular point space.
Then $T$ is weakly countably compact. | Let $T = \struct {S, \tau_p}$ be a finite particular point space.
By definition, $T$ is a finite topological space.
The result follows from Finite Space Satisfies All Compactness Properties.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Finite Particular Point Space|finite particular point space]].
Then $T$ is [[Definition:Weakly Countably Compact Space|weakly countably compact]]. | Let $T = \struct {S, \tau_p}$ be a [[Definition:Finite Particular Point Space|finite particular point space]].
By definition, $T$ is a [[Definition:Finite Topological Space|finite topological space]].
The result follows from [[Finite Space Satisfies All Compactness Properties]].
{{qed}} | Finite Particular Point Space is Weakly Countably Compact/Proof 1 | https://proofwiki.org/wiki/Finite_Particular_Point_Space_is_Weakly_Countably_Compact | https://proofwiki.org/wiki/Finite_Particular_Point_Space_is_Weakly_Countably_Compact/Proof_1 | [
"Finite Particular Point Space is Weakly Countably Compact",
"Finite Particular Point Topologies",
"Examples of Weakly Countably Compact Spaces"
] | [
"Definition:Particular Point Topology/Finite",
"Definition:Weakly Countably Compact Space"
] | [
"Definition:Particular Point Topology/Finite",
"Definition:Finite Topological Space",
"Finite Space Satisfies All Compactness Properties"
] |
proofwiki-23426 | Finite Particular Point Space is Compact | Let $T = \struct {S, \tau_p}$ be a finite particular point space.
Then $T$ is compact. | By definition, $T$ is a finite topological space.
The result follows from Finite Topological Space is Compact.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Finite Particular Point Space|finite particular point space]].
Then $T$ is [[Definition:Compact Topological Space| compact]]. | By definition, $T$ is a [[Definition:Finite Topological Space|finite topological space]].
The result follows from [[Finite Topological Space is Compact]].
{{qed}} | Finite Particular Point Space is Compact | https://proofwiki.org/wiki/Finite_Particular_Point_Space_is_Compact | https://proofwiki.org/wiki/Finite_Particular_Point_Space_is_Compact | [
"Finite Particular Point Topologies",
"Examples of Compact Topological Spaces"
] | [
"Definition:Particular Point Topology/Finite",
"Definition:Compact Topological Space"
] | [
"Definition:Finite Topological Space",
"Finite Topological Space is Compact"
] |
proofwiki-23427 | Open Extension of Non-Trivial Particular Point Space is T0 | $T^*_{\bar q}$ is a $T_0$ space. | We have that a Particular Point Space is $T_0$.
Then from Condition for Open Extension Space to be $T_0$, it follows that $T^*_{\bar q}$ is a $T_0$ space.
{{qed}} | $T^*_{\bar q}$ is a [[Definition:T0 Space|$T_0$ space]]. | We have that a [[Particular Point Space is T0|Particular Point Space is $T_0$]].
Then from [[Condition for Open Extension Space to be T0|Condition for Open Extension Space to be $T_0$]], it follows that $T^*_{\bar q}$ is a [[Definition:T0 Space|$T_0$ space]].
{{qed}} | Open Extension of Non-Trivial Particular Point Space is T0 | https://proofwiki.org/wiki/Open_Extension_of_Non-Trivial_Particular_Point_Space_is_T0 | https://proofwiki.org/wiki/Open_Extension_of_Non-Trivial_Particular_Point_Space_is_T0 | [
"Open Extension Topologies",
"Particular Point Topologies",
"Examples of T0 Spaces"
] | [
"Definition:T0 Space"
] | [
"Particular Point Space is T0",
"Condition for Open Extension Space to be T0",
"Definition:T0 Space"
] |
proofwiki-23428 | Open Extension of Non-Trivial Particular Point Space is T4 | $T^*_{\bar q}$ is a $T_4$ space. | We have directly that an Open Extension Topology is $T_4$.
{{qed}} | $T^*_{\bar q}$ is a [[Definition:T4 Space|$T_4$ space]]. | We have directly that an [[Open Extension Topology is T4|Open Extension Topology is $T_4$]].
{{qed}} | Open Extension of Non-Trivial Particular Point Space is T4 | https://proofwiki.org/wiki/Open_Extension_of_Non-Trivial_Particular_Point_Space_is_T4 | https://proofwiki.org/wiki/Open_Extension_of_Non-Trivial_Particular_Point_Space_is_T4 | [
"Open Extension Topologies",
"Particular Point Topologies",
"Examples of T4 Spaces"
] | [
"Definition:T4 Space"
] | [
"Open Extension Topology is T4"
] |
proofwiki-23429 | Open Extension of Non-Trivial Particular Point Space is not T1 | $T^*_{\bar q}$ is not a $T_1$ space. | We have directly that an Open Extension Topology is not $T_1$.
{{qed}} | $T^*_{\bar q}$ is not a [[Definition:T1 Space|$T_1$ space]]. | We have directly that an [[Open Extension Topology is not T1|Open Extension Topology is not $T_1$]].
{{qed}} | Open Extension of Non-Trivial Particular Point Space is not T1 | https://proofwiki.org/wiki/Open_Extension_of_Non-Trivial_Particular_Point_Space_is_not_T1 | https://proofwiki.org/wiki/Open_Extension_of_Non-Trivial_Particular_Point_Space_is_not_T1 | [
"Open Extension Topologies",
"Particular Point Topologies",
"Examples of T1 Spaces"
] | [
"Definition:T1 Space"
] | [
"Open Extension Topology is not T1"
] |
proofwiki-23430 | Open Extension of Non-Trivial Particular Point Space is not T5 | $T^*_{\bar q}$ is not a $T_5$ space. | A Particular Point Topology with three points or more is not $T_4$.
From $T_5$ Space is $T_4$, it follows that $T$ is not a $T_5$ space.
It follows from Condition for Open Extension Space to be $T_5$ that $T^*_{\bar q}$ is not a $T_5$ space.
{{qed}} | $T^*_{\bar q}$ is not a [[Definition:T5 Space|$T_5$ space]]. | A [[Particular Point Topology with Three Points is not T4|Particular Point Topology with three points or more is not $T_4$]].
From [[T5 Space is T4|$T_5$ Space is $T_4$]], it follows that $T$ is not a [[Definition:T5 Space|$T_5$ space]].
It follows from [[Condition for Open Extension Space to be T5|Condition for Open... | Open Extension of Non-Trivial Particular Point Space is not T5 | https://proofwiki.org/wiki/Open_Extension_of_Non-Trivial_Particular_Point_Space_is_not_T5 | https://proofwiki.org/wiki/Open_Extension_of_Non-Trivial_Particular_Point_Space_is_not_T5 | [
"Open Extension Topologies",
"Particular Point Topologies",
"Examples of T5 Spaces"
] | [
"Definition:T5 Space"
] | [
"Particular Point Topology with Three Points is not T4",
"T5 Space is T4",
"Definition:T5 Space",
"Condition for Open Extension Space to be T5",
"Definition:T5 Space"
] |
proofwiki-23431 | Product Formula for Tangent | :$\ds \map \tan {n z} = \begin{cases} \paren {-1}^r \ds \prod_{k \mathop = 0}^{n - 1} \dfrac {\map \sin {z + \dfrac {k \pi} n} } {\map \cos {z + \dfrac {\paren {2 k + 1 } \pi} {2 n} } } & : n = 2 r & r \in \N \\ \paren {-1}^r \ds \prod_{k \mathop = 0}^{n - 1} \map \tan {z + \dfrac {k \pi} n} & : n = 2 r + 1 & r \in \N ... | From Product Formula for Sine, we have:
:$\ds \map \sin {n z} = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \frac {k \pi} n}$
From Product Formula for Cosine Lemma 1, we have:
When $n \equiv 1 \pmod 4$
Then:
:$\ds \map \cos {n z} = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \cos {z + \dfrac {2 k \pi} {2 n... | :$\ds \map \tan {n z} = \begin{cases} \paren {-1}^r \ds \prod_{k \mathop = 0}^{n - 1} \dfrac {\map \sin {z + \dfrac {k \pi} n} } {\map \cos {z + \dfrac {\paren {2 k + 1 } \pi} {2 n} } } & : n = 2 r & r \in \N \\ \paren {-1}^r \ds \prod_{k \mathop = 0}^{n - 1} \map \tan {z + \dfrac {k \pi} n} & : n = 2 r + 1 & r \in \N ... | From [[Product Formula for Sine]], we have:
:$\ds \map \sin {n z} = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \frac {k \pi} n}$
From [[Product Formula for Cosine/Lemma 1|Product Formula for Cosine Lemma 1]], we have:
When $n \equiv 1 \pmod 4$
Then:
:$\ds \map \cos {n z} = 2^{n - 1} \prod_{k \mathop =... | Product Formula for Tangent | https://proofwiki.org/wiki/Product_Formula_for_Tangent | https://proofwiki.org/wiki/Product_Formula_for_Tangent | [
"Product Formula for Tangent",
"Tangent Function"
] | [] | [
"Product Formula for Sine",
"Product Formula for Cosine/Lemma 1",
"Product Formula for Cosine/Lemma 2",
"Definition:Odd Integer",
"Product Formula for Cosine/Lemma 3",
"Product Formula for Cosine/Lemma 4",
"Definition:Even Integer"
] |
proofwiki-23432 | Countable Complement Space is not T3 | Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$.
Then $T$ is not a $T_3$ space. | Let $S$ be an uncountable set.
Let $T = \struct {S, \tau}$ be the countable complement space on $S$.
We have that:
:Countable Complement Space is a $T_1$ space
From $T_1$ Space is $T_0$ Space, $T$ is a $T_0$ space.
{{AimForCont}} $T$ is a $T_3$ space.
From Space which is $T_3$ and $T_0$ is also $T_2$, $T$ is a $T_2$ sp... | Let $T = \struct {S, \tau}$ be a [[Definition:Countable Complement Topology|countable complement topology]] on an [[Definition:Uncountable Set|uncountable set]] $S$.
Then $T$ is not a [[Definition:T3 Space|$T_3$ space]]. | Let $S$ be an [[Definition:Uncountable Set|uncountable set]].
Let $T = \struct {S, \tau}$ be the [[Definition:Countable Complement Space|countable complement space]] on $S$.
We have that:
:[[Countable Complement Space is T1|Countable Complement Space is a $T_1$ space]]
From [[T1 Space is T0 Space|$T_1$ Space is $T_0... | Countable Complement Space is not T3 | https://proofwiki.org/wiki/Countable_Complement_Space_is_not_T3 | https://proofwiki.org/wiki/Countable_Complement_Space_is_not_T3 | [
"Countable Complement Topologies",
"Examples of T3 Spaces"
] | [
"Definition:Countable Complement Topology",
"Definition:Uncountable/Set",
"Definition:T3 Space"
] | [
"Definition:Uncountable/Set",
"Definition:Countable Complement Topology",
"Countable Complement Space is T1",
"T1 Space is T0",
"Definition:T0 Space",
"Definition:T3 Space",
"Space which is T3 and T0 is also T2",
"Definition:T2 Space",
"Countable Complement Space is not T2",
"Definition:T2 Space",... |
proofwiki-23433 | Countable Complement Space is not T4 | Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$.
Then $T$ is not a $T_4$ space. | We have that a Normal Space is $T_3$.
But from Countable Complement Space is not $T_3$, $T$ cannot be a normal space.
By definition, a normal space is a space that is both a $T_1$ space and a $T_4$ space.
But $T$ is a $T_1$ space and not a normal space.
So it follows that $T$ cannot be a $T_4$ space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Countable Complement Topology|countable complement topology]] on an [[Definition:Uncountable Set|uncountable set]] $S$.
Then $T$ is not a [[Definition:T4 Space|$T_4$ space]]. | We have that a [[Normal Space is T3|Normal Space is $T_3$]].
But from [[Countable Complement Space is not T3|Countable Complement Space is not $T_3$]], $T$ cannot be a [[Definition:Normal Space|normal space]].
By definition, a [[Definition:Normal Space|normal space]] is a space that is both a [[Definition:T1 Space|$T... | Countable Complement Space is not T4 | https://proofwiki.org/wiki/Countable_Complement_Space_is_not_T4 | https://proofwiki.org/wiki/Countable_Complement_Space_is_not_T4 | [
"Countable Complement Topologies",
"Examples of T4 Spaces"
] | [
"Definition:Countable Complement Topology",
"Definition:Uncountable/Set",
"Definition:T4 Space"
] | [
"Normal Space is T3",
"Countable Complement Space is not T3",
"Definition:Normal Space",
"Definition:Normal Space",
"Definition:T1 Space",
"Definition:T4 Space",
"Definition:T1 Space",
"Definition:Normal Space",
"Definition:T4 Space"
] |
proofwiki-23434 | Countable Complement Space is not T5 | Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$.
Then $T$ is not a $T_5$ space. | A $T_5$ Space is $T_4$.
But from Countable Complement Space is not $T_4$, $T$ cannot be a $T_5$ space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Countable Complement Topology|countable complement topology]] on an [[Definition:Uncountable Set|uncountable set]] $S$.
Then $T$ is not a [[Definition:T5 Space|$T_5$ space]]. | A [[T5 Space is T4|$T_5$ Space is $T_4$]].
But from [[Countable Complement Space is not T4|Countable Complement Space is not $T_4$]], $T$ cannot be a [[Definition:T5 Space|$T_5$ space]].
{{qed}} | Countable Complement Space is not T5 | https://proofwiki.org/wiki/Countable_Complement_Space_is_not_T5 | https://proofwiki.org/wiki/Countable_Complement_Space_is_not_T5 | [
"Countable Complement Topologies",
"Examples of T5 Spaces"
] | [
"Definition:Countable Complement Topology",
"Definition:Uncountable/Set",
"Definition:T5 Space"
] | [
"T5 Space is T4",
"Countable Complement Space is not T4",
"Definition:T5 Space"
] |
proofwiki-23435 | Compact Complement Space is not T2 | Let $T = \struct {\R, \tau}$ be the compact complement space on $\R$.
Then $T$ is not a $T_2$ (Hausdorff) space. | We have:
:Compact Complement Topology is Irreducible
:Irreducible Hausdorff Space is Singleton
Hence the result.
{{qed}} | Let $T = \struct {\R, \tau}$ be the [[Definition:Compact Complement Space|compact complement space]] on $\R$.
Then $T$ is not a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | We have:
:[[Compact Complement Topology is Irreducible]]
:[[Irreducible Hausdorff Space is Singleton]]
Hence the result.
{{qed}} | Compact Complement Space is not T2 | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_T2 | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_T2 | [
"Compact Complement Topology",
"Examples of Hausdorff Spaces"
] | [
"Definition:Compact Complement Topology",
"Definition:T2 Space"
] | [
"Compact Complement Topology is Irreducible",
"Irreducible Hausdorff Space is Singleton"
] |
proofwiki-23436 | Compact Complement Space is not T2.5 | Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$.
Then $T$ is not a $T_{2 \frac 1 2}$ space. | From Compact Complement Space is not $T_2$, $T$ is not a $T_2$ space.
So from $T_{2 \frac 1 2}$ Space is $T_2$, it cannot be the case that $T$ is a $T_{2 \frac 1 2}$ space.
{{qed}} | Let $T = \struct {\R, \tau}$ be the [[Definition:Compact Complement Topology|compact complement topology]] on $\R$.
Then $T$ is not a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]]. | From [[Compact Complement Space is not T2|Compact Complement Space is not $T_2$]], $T$ is not a [[Definition:T2 Space|$T_2$ space]].
So from [[T2.5 Space is T2|$T_{2 \frac 1 2}$ Space is $T_2$]], it cannot be the case that $T$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]].
{{qed}} | Compact Complement Space is not T2.5 | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_T2.5 | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_T2.5 | [
"Compact Complement Topology",
"Examples of T2.5 Spaces"
] | [
"Definition:Compact Complement Topology",
"Definition:T2.5 Space"
] | [
"Compact Complement Space is not T2",
"Definition:T2 Space",
"T2.5 Space is T2",
"Definition:T2.5 Space"
] |
proofwiki-23437 | Compact Complement Space is not T3 | Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$.
Then $T$ is not a $T_3$ space. | We have that a Compact Complement Space is a $T_0$ space.
From Compact Complement Space is not Regular, $T$ is not a regular space.
By definition, a regular space is a space that is both a $T_0$ space and a $T_3$ space.
But $T$ is a $T_0$ space and not a regular space.
So it follows that $T$ cannot be a $T_3$ space.
{{... | Let $T = \struct {\R, \tau}$ be the [[Definition:Compact Complement Topology|compact complement topology]] on $\R$.
Then $T$ is not a [[Definition:T3 Space|$T_3$ space]]. | We have that a [[Compact Complement Space is T0|Compact Complement Space is a $T_0$ space]].
From [[Compact Complement Space is not Regular]], $T$ is not a [[Definition:Regular Space|regular space]].
By definition, a [[Definition:Regular Space|regular space]] is a space that is both a [[Definition:T0 Space|$T_0$ spa... | Compact Complement Space is not T3 | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_T3 | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_T3 | [
"Compact Complement Topology",
"Examples of T3 Spaces"
] | [
"Definition:Compact Complement Topology",
"Definition:T3 Space"
] | [
"Compact Complement Space is T0",
"Compact Complement Space is not Regular",
"Definition:Regular Space",
"Definition:Regular Space",
"Definition:T0 Space",
"Definition:T3 Space",
"Definition:T0 Space",
"Definition:Regular Space",
"Definition:T3 Space"
] |
proofwiki-23438 | Compact Complement Space is not T4 | Let $T = \struct {\R, \tau}$ be the compact complement space on $\R$.
Then $T$ is not a $T_4$ space. | We have that a Compact Complement Space is a $T_1$ space.
By definition, a normal space is a space that is both a $T_1$ space and a $T_4$ space.
But from Compact Complement Space is not Normal, $T$ is not a normal space.
So it follows that $T$ cannot be a $T_4$ space.
{{qed}} | Let $T = \struct {\R, \tau}$ be the [[Definition:Compact Complement Space|compact complement space]] on $\R$.
Then $T$ is not a [[Definition:T4 Space|$T_4$ space]]. | We have that a [[Compact Complement Space is T1|Compact Complement Space is a $T_1$ space]].
By definition, a [[Definition:Normal Space|normal space]] is a space that is both a [[Definition:T1 Space|$T_1$ space]] and a [[Definition:T4 Space|$T_4$ space]].
But from [[Compact Complement Space is not Normal]], $T$ is no... | Compact Complement Space is not T4 | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_T4 | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_T4 | [
"Compact Complement Topology",
"Examples of T4 Spaces"
] | [
"Definition:Compact Complement Topology",
"Definition:T4 Space"
] | [
"Compact Complement Space is T1",
"Definition:Normal Space",
"Definition:T1 Space",
"Definition:T4 Space",
"Compact Complement Space is not Normal",
"Definition:Normal Space",
"Definition:T4 Space"
] |
proofwiki-23439 | Compact Complement Space is not T5 | Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$.
Then $T$ is not a $T_5$ space. | We have that a $T_5$ Space is $T_4$.
We have that a Compact Complement Space is not a $T_4$ space.
So as $T$ is not a $T_4$ space, $T$ cannot be a $T_5$ space.
{{qed}} | Let $T = \struct {\R, \tau}$ be the [[Definition:Compact Complement Topology|compact complement topology]] on $\R$.
Then $T$ is not a [[Definition:T5 Space|$T_5$ space]]. | We have that a [[T5 Space is T4|$T_5$ Space is $T_4$]].
We have that a [[Compact Complement Space is not T4|Compact Complement Space is not a $T_4$ space]].
So as $T$ is not a [[Definition:T4 Space|$T_4$ space]], $T$ cannot be a [[Definition:T5 Space|$T_5$ space]].
{{qed}} | Compact Complement Space is not T5 | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_T5 | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_T5 | [
"Compact Complement Topology",
"Examples of T5 Spaces"
] | [
"Definition:Compact Complement Topology",
"Definition:T5 Space"
] | [
"T5 Space is T4",
"Compact Complement Space is not T4",
"Definition:T4 Space",
"Definition:T5 Space"
] |
proofwiki-23440 | Compact Complement Space is T0 | Let $T = \struct {\R, \tau}$ be the compact complement space on $\R$.
Then $T$ is a $T_0$ space. | We have that a Compact Complement Space is a $T_1$ space.
From $T_1$ Space is $T_0$ Space, $T$ is a $T_0$ space.
{{qed}}
Category:Compact Complement Topology
Category:Examples of T0 Spaces
k6axht5asqex6nixrgzyj34qus5xkjt | Let $T = \struct {\R, \tau}$ be the [[Definition:Compact Complement Space |compact complement space]] on $\R$.
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | We have that a [[Compact Complement Space is T1|Compact Complement Space is a $T_1$ space]].
From [[T1 Space is T0 Space|$T_1$ Space is $T_0$ Space]], $T$ is a [[Definition:T0 Space|$T_0$ space]].
{{qed}}
[[Category:Compact Complement Topology]]
[[Category:Examples of T0 Spaces]]
k6axht5asqex6nixrgzyj34qus5xkjt | Compact Complement Space is T0 | https://proofwiki.org/wiki/Compact_Complement_Space_is_T0 | https://proofwiki.org/wiki/Compact_Complement_Space_is_T0 | [
"Compact Complement Topology",
"Examples of T0 Spaces"
] | [
"Definition:Compact Complement Space ",
"Definition:T0 Space"
] | [
"Compact Complement Space is T1",
"T1 Space is T0",
"Definition:T0 Space",
"Category:Compact Complement Topology",
"Category:Examples of T0 Spaces"
] |
proofwiki-23441 | Compact Complement Space is not Regular | Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$.
Then $T$ is not a regular space. | {{Recall|Regular Space|regular space}}
{{:Definition:Regular Space/Definition 3}}
From Compact Complement Space is not $T_2$, $T$ is not a $T_2$ space.
Hence the result.
{{qed}}
Category:Compact Complement Topology
Category:Examples of Regular Spaces
s6qbfmaxfikqy5bz0usq3hzne4vsxb4 | Let $T = \struct {\R, \tau}$ be the [[Definition:Compact Complement Topology|compact complement topology]] on $\R$.
Then $T$ is not a [[Definition:Regular Space|regular space]]. | {{Recall|Regular Space|regular space}}
{{:Definition:Regular Space/Definition 3}}
From [[Compact Complement Space is not T2|Compact Complement Space is not $T_2$]], $T$ is not a [[Definition:T2 Space|$T_2$ space]].
Hence the result.
{{qed}}
[[Category:Compact Complement Topology]]
[[Category:Examples of Regular Spac... | Compact Complement Space is not Regular | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_Regular | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_Regular | [
"Compact Complement Topology",
"Examples of Regular Spaces"
] | [
"Definition:Compact Complement Topology",
"Definition:Regular Space"
] | [
"Compact Complement Space is not T2",
"Definition:T2 Space",
"Category:Compact Complement Topology",
"Category:Examples of Regular Spaces"
] |
proofwiki-23442 | Compact Complement Space is not Normal | Let $T = \struct {\R, \tau}$ be the compact complement space on $\R$.
Then $T$ is not a normal space. | We have that a Compact Complement Space is $T_1$.
We also have that a Normal Space is a $T_3$.
But as $T$ is not a $T_3$ space, $T$ cannot be a normal space.
{{qed}}
Category:Compact Complement Topology
Category:Examples of Normal Spaces
07eqsqkdo2k6qcbh41pdeu1rm469s49 | Let $T = \struct {\R, \tau}$ be the [[Definition:Compact Complement Space|compact complement space]] on $\R$.
Then $T$ is not a [[Definition:Normal Space|normal space]]. | We have that a [[Compact Complement Space is T1|Compact Complement Space is $T_1$]].
We also have that a [[Normal Space is T3|Normal Space is a $T_3$]].
But as $T$ is not a [[Definition:T3 Space|$T_3$ space]], $T$ cannot be a [[Definition:Normal Space|normal space]].
{{qed}}
[[Category:Compact Complement Topology]]
... | Compact Complement Space is not Normal | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_Normal | https://proofwiki.org/wiki/Compact_Complement_Space_is_not_Normal | [
"Compact Complement Topology",
"Examples of Normal Spaces"
] | [
"Definition:Compact Complement Topology",
"Definition:Normal Space"
] | [
"Compact Complement Space is T1",
"Normal Space is T3",
"Definition:T3 Space",
"Definition:Normal Space",
"Category:Compact Complement Topology",
"Category:Examples of Normal Spaces"
] |
proofwiki-23443 | Similar Matrices have Same Characteristic Polynomial | Let $\mathbf A$ and $\mathbf B$ be similar $n$ by $n$ square matrices.
Then $\mathbf A$ and $\mathbf B$ have the same characteristic polynomial. | Because $\mathbf A$ and $\mathbf B$ are similar, there exists an invertible matrix $\mathbf P$ such that:
:$\mathbf A = \mathbf P \mathbf B \mathbf P^{-1}$
Then:
{{begin-eqn}}
{{eqn | l = \map \det {X\mathbf I_n - \mathbf A}
| r = \map \det {X \mathbf I_n - \mathbf P \mathbf B \mathbf P^{-1} }
}}
{{eqn | r = \map... | Let $\mathbf A$ and $\mathbf B$ be [[Definition:Matrix Similarity|similar]] $n$ by $n$ [[Definition:Square Matrix|square matrices]].
Then $\mathbf A$ and $\mathbf B$ have the same [[Definition:Characteristic Polynomial of Matrix|characteristic polynomial]]. | Because $\mathbf A$ and $\mathbf B$ are [[Definition:Matrix Similarity|similar]], there exists an [[Definition:Nonsingular Matrix|invertible matrix]] $\mathbf P$ such that:
:$\mathbf A = \mathbf P \mathbf B \mathbf P^{-1}$
Then:
{{begin-eqn}}
{{eqn | l = \map \det {X\mathbf I_n - \mathbf A}
| r = \map \det {X \... | Similar Matrices have Same Characteristic Polynomial | https://proofwiki.org/wiki/Similar_Matrices_have_Same_Characteristic_Polynomial | https://proofwiki.org/wiki/Similar_Matrices_have_Same_Characteristic_Polynomial | [
"Characteristic Polynomial of Matrix",
"Matrix Similarity"
] | [
"Definition:Matrix Similarity",
"Definition:Matrix/Square Matrix",
"Definition:Characteristic Polynomial of Matrix"
] | [
"Definition:Matrix Similarity",
"Definition:Nonsingular Matrix",
"Determinant of Matrix Product",
"Determinant of Inverse Matrix",
"Category:Characteristic Polynomial of Matrix",
"Category:Matrix Similarity"
] |
proofwiki-23444 | Limit of Image of Sequence/Topology | Let $\struct {X, \tau_X}$ and $\struct {Y, \tau_Y}$ be topological spaces.
Let $f: X \to Y$ be a continuous at (the point) $x$.
Let $\sequence {x_n}$ be a convergent sequence to the limit $x$.
Then the sequence $\sequence {\map f {x_n}}$ is convergent to the limit $f(x)$. | Let $V$ be an arbitrary open neighborhood of $\map f x$.
By continuous at (the point) $f(x)$ there exists an open neighborhood $U$ of $x$ such that $f[U] \subseteq V$.
By convergence of $\sequence {x_n}$ to $x$ there exists $N \in \R_{\ge 0}$ such that:
:$\forall n \in \N : n \ge N \implies x_n \in U$
Thus we found $... | Let $\struct {X, \tau_X}$ and $\struct {Y, \tau_Y}$ be [[Definition:Topological Space|topological spaces]].
Let $f: X \to Y$ be a [[Definition:Continuous Mapping (Topology)/Point|continuous at (the point)]] $x$.
Let $\sequence {x_n}$ be a [[Definition:Convergent Sequence/Topology|convergent sequence]] to the [[Defin... | Let $V$ be an arbitrary [[Definition:Open Neighborhood|open neighborhood]] of $\map f x$.
By [[Definition:Continuous Mapping (Topology)/Point|continuous at (the point)]] $f(x)$ there exists an [[Definition:Open Neighborhood|open neighborhood]] $U$ of $x$ such that $f[U] \subseteq V$.
By [[Definition:Convergent Sequ... | Limit of Image of Sequence/Topology | https://proofwiki.org/wiki/Limit_of_Image_of_Sequence/Topology | https://proofwiki.org/wiki/Limit_of_Image_of_Sequence/Topology | [
"Continuous Mappings (Topology)",
"Convergent Sequences (Topology)"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Point",
"Definition:Convergent Sequence/Topology",
"Definition:Limit of Sequence/Topological Space",
"Definition:Convergent Sequence/Topology",
"Definition:Limit of Sequence/Topological Space"
] | [
"Definition:Open Neighborhood",
"Definition:Continuous Mapping (Topology)/Point",
"Definition:Open Neighborhood",
"Definition:Convergent Sequence/Topology",
"Definition:Open Neighborhood",
"Definition:Convergent Sequence/Topology",
"Definition:Limit of Sequence/Topological Space",
"Category:Continuous... |
proofwiki-23445 | Fort Space is T2 | Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$.
Then $T$ is a $T_2$ (Hausdorff) space. | From Fort Space is $T_{2 \frac 1 2}$, $T$ is a $T_{2 \frac 1 2}$ space.
The result follows from $T_{2 \frac 1 2}$ Space is $T_2$.
{{qed}}
Category:Fort Spaces
Category:Examples of Hausdorff Spaces
9fjpg9f1w0v7ski4n342pqa6yyjam81 | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fort Space|Fort space]] on an [[Definition:Infinite Set|infinite set]] $S$.
Then $T$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | From [[Fort Space is T2.5|Fort Space is $T_{2 \frac 1 2}$]], $T$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]].
The result follows from [[T2.5 Space is T2|$T_{2 \frac 1 2}$ Space is $T_2$]].
{{qed}}
[[Category:Fort Spaces]]
[[Category:Examples of Hausdorff Spaces]]
9fjpg9f1w0v7ski4n342pqa6yyjam81 | Fort Space is T2 | https://proofwiki.org/wiki/Fort_Space_is_T2 | https://proofwiki.org/wiki/Fort_Space_is_T2 | [
"Fort Spaces",
"Examples of Hausdorff Spaces"
] | [
"Definition:Fort Space",
"Definition:Infinite Set",
"Definition:T2 Space"
] | [
"Fort Space is T2.5",
"Definition:T2.5 Space",
"T2.5 Space is T2",
"Category:Fort Spaces",
"Category:Examples of Hausdorff Spaces"
] |
proofwiki-23446 | Fort Space is T2.5 | Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$.
Then $T$ is a $T_{2 \frac 1 2}$ space. | From Fort Space is Regular, $T$ is a regular space.
The result follows from Regular Space is $T_{2 \frac 1 2}$.
{{qed}}
Category:Fort Spaces
Category:Examples of T2.5 Spaces
0wgyalzvggdh0itwu9whh42lhw2mald | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fort Space|Fort space]] on an [[Definition:Infinite Set|infinite set]] $S$.
Then $T$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]]. | From [[Fort Space is Regular]], $T$ is a [[Definition:Regular Space|regular space]].
The result follows from [[Regular Space is T2.5|Regular Space is $T_{2 \frac 1 2}$]].
{{qed}}
[[Category:Fort Spaces]]
[[Category:Examples of T2.5 Spaces]]
0wgyalzvggdh0itwu9whh42lhw2mald | Fort Space is T2.5 | https://proofwiki.org/wiki/Fort_Space_is_T2.5 | https://proofwiki.org/wiki/Fort_Space_is_T2.5 | [
"Fort Spaces",
"Examples of T2.5 Spaces"
] | [
"Definition:Fort Space",
"Definition:Infinite Set",
"Definition:T2.5 Space"
] | [
"Fort Space is Regular",
"Definition:Regular Space",
"Regular Space is T2.5",
"Category:Fort Spaces",
"Category:Examples of T2.5 Spaces"
] |
proofwiki-23447 | Fortissimo Space is T0 | Let $T = \struct {S, \tau_p}$ be a Fortissimo space.
Then $T$ is a $T_0$ space. | Follows directly from:
:Fortissimo Space is $T_1$
:$T_1$ Space is $T_0$ Space
{{qed}}
Category:Fortissimo Spaces
Category:Examples of T0 Spaces
5cm121tewbg0neuhbfyss3rgsuaxa93 | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fortissimo Space|Fortissimo space]].
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | Follows directly from:
:[[Fortissimo Space is T1|Fortissimo Space is $T_1$]]
:[[T1 Space is T0 Space|$T_1$ Space is $T_0$ Space]]
{{qed}}
[[Category:Fortissimo Spaces]]
[[Category:Examples of T0 Spaces]]
5cm121tewbg0neuhbfyss3rgsuaxa93 | Fortissimo Space is T0 | https://proofwiki.org/wiki/Fortissimo_Space_is_T0 | https://proofwiki.org/wiki/Fortissimo_Space_is_T0 | [
"Fortissimo Spaces",
"Examples of T0 Spaces",
"Fortissimo Spaces",
"Examples of T0 Spaces"
] | [
"Definition:Fortissimo Space",
"Definition:T0 Space"
] | [
"Fortissimo Space is T1",
"T1 Space is T0",
"Category:Fortissimo Spaces",
"Category:Examples of T0 Spaces"
] |
proofwiki-23448 | Arens-Fort Space is T2 | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a $T_2$ (Hausdorff) space. | We have:
:Fort Space is $T_2$
:Arens-Fort Space is Expansion of Fort Space.
The result follows from $T_2$ Property is Preserved under Expansion.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | We have:
:[[Fort Space is T2|Fort Space is $T_2$]]
:[[Arens-Fort Space is Expansion of Fort Space]].
The result follows from [[T2 Property is Preserved under Expansion|$T_2$ Property is Preserved under Expansion]].
{{qed}} | Arens-Fort Space is T2 | https://proofwiki.org/wiki/Arens-Fort_Space_is_T2 | https://proofwiki.org/wiki/Arens-Fort_Space_is_T2 | [
"Arens-Fort Space",
"Examples of Hausdorff Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:T2 Space"
] | [
"Fort Space is T2",
"Arens-Fort Space is Expansion of Countable Fort Space",
"T2 Property is Preserved under Expansion"
] |
proofwiki-23449 | Arens-Fort Space is T0 | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a $T_0$ space. | We have that the Arens-Fort Space is $T_1$.
The result follows from $T_1$ Space is $T_0$ Space.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | We have that the [[Arens-Fort Space is T1|Arens-Fort Space is $T_1$]].
The result follows from [[T1 Space is T0 Space|$T_1$ Space is $T_0$ Space]].
{{qed}} | Arens-Fort Space is T0 | https://proofwiki.org/wiki/Arens-Fort_Space_is_T0 | https://proofwiki.org/wiki/Arens-Fort_Space_is_T0 | [
"Arens-Fort Space",
"Examples of T0 Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:T0 Space"
] | [
"Arens-Fort Space is T1",
"T1 Space is T0"
] |
proofwiki-23450 | Arens-Fort Space is Normal | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a normal space. | We have:
:Arens-Fort Space is Completely Normal
:Completely Normal Space is Normal.
{{qed}}
Category:Arens-Fort Space
Category:Examples of Normal Spaces
pyxfo8cg9hfk5ba8m82j72uyjz3nwo8 | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:Normal Space|normal space]]. | We have:
:[[Arens-Fort Space is Completely Normal]]
:[[Completely Normal Space is Normal]].
{{qed}}
[[Category:Arens-Fort Space]]
[[Category:Examples of Normal Spaces]]
pyxfo8cg9hfk5ba8m82j72uyjz3nwo8 | Arens-Fort Space is Normal | https://proofwiki.org/wiki/Arens-Fort_Space_is_Normal | https://proofwiki.org/wiki/Arens-Fort_Space_is_Normal | [
"Arens-Fort Space",
"Examples of Normal Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Normal Space"
] | [
"Arens-Fort Space is Completely Normal",
"Completely Normal Space is Normal",
"Category:Arens-Fort Space",
"Category:Examples of Normal Spaces"
] |
proofwiki-23451 | Arens-Fort Space is Completely Regular | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a completely regular space. | We have:
:Arens-Fort Space is Normal
:Normal Space is Completely Regular.
{{qed}}
Category:Arens-Fort Space
Category:Examples of Completely Regular Spaces
n4furyqhdxto1ta6gel0rnbcnk1yryt | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:Completely Regular Space|completely regular space]]. | We have:
:[[Arens-Fort Space is Normal]]
:[[Normal Space is Completely Regular]].
{{qed}}
[[Category:Arens-Fort Space]]
[[Category:Examples of Completely Regular Spaces]]
n4furyqhdxto1ta6gel0rnbcnk1yryt | Arens-Fort Space is Completely Regular | https://proofwiki.org/wiki/Arens-Fort_Space_is_Completely_Regular | https://proofwiki.org/wiki/Arens-Fort_Space_is_Completely_Regular | [
"Arens-Fort Space",
"Examples of Completely Regular Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Completely Regular Space"
] | [
"Arens-Fort Space is Normal",
"Normal Space is Completely Regular",
"Category:Arens-Fort Space",
"Category:Examples of Completely Regular Spaces"
] |
proofwiki-23452 | Arens-Fort Space is Regular | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a regular space. | We have:
:Arens-Fort Space is Completely Regular
:Completely Regular Space is Regular.
{{qed}}
Category:Arens-Fort Space
Category:Examples of Regular Spaces
42bc3s0yp8kh766eb639y64ty8dbhlf | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:Regular Space|regular space]]. | We have:
:[[Arens-Fort Space is Completely Regular]]
:[[Completely Regular Space is Regular]].
{{qed}}
[[Category:Arens-Fort Space]]
[[Category:Examples of Regular Spaces]]
42bc3s0yp8kh766eb639y64ty8dbhlf | Arens-Fort Space is Regular | https://proofwiki.org/wiki/Arens-Fort_Space_is_Regular | https://proofwiki.org/wiki/Arens-Fort_Space_is_Regular | [
"Arens-Fort Space",
"Examples of Regular Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Regular Space"
] | [
"Arens-Fort Space is Completely Regular",
"Completely Regular Space is Regular",
"Category:Arens-Fort Space",
"Category:Examples of Regular Spaces"
] |
proofwiki-23453 | Arens-Fort Space is T3 | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a $T_3$ space. | We have Arens-Fort Space is Regular.
{{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
The result follows.
{{qed}}
Category:Arens-Fort Space
Category:Examples of T3 Spaces
67mr12ufmqugzk5sfcenaz07z1lpij5 | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:T3 Space|$T_3$ space]]. | We have [[Arens-Fort Space is Regular]].
{{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
The result follows.
{{qed}}
[[Category:Arens-Fort Space]]
[[Category:Examples of T3 Spaces]]
67mr12ufmqugzk5sfcenaz07z1lpij5 | Arens-Fort Space is T3 | https://proofwiki.org/wiki/Arens-Fort_Space_is_T3 | https://proofwiki.org/wiki/Arens-Fort_Space_is_T3 | [
"Arens-Fort Space",
"Examples of T3 Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:T3 Space"
] | [
"Arens-Fort Space is Regular",
"Category:Arens-Fort Space",
"Category:Examples of T3 Spaces"
] |
proofwiki-23454 | Modified Fort Space is T0 | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is a $T_0$ space. | We have that a Modified Fort Space is $T_1$.
From $T_1$ Space is $T_0$ Space, $T$ is a $T_0$ space.
{{qed}}
Category:Modified Fort Spaces
Category:Examples of T0 Spaces
jcwy2h4yjjkh5rx6s4zac16xosmdwgw | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | We have that a [[Modified Fort Space is T1|Modified Fort Space is $T_1$]].
From [[T1 Space is T0 Space|$T_1$ Space is $T_0$ Space]], $T$ is a [[Definition:T0 Space|$T_0$ space]].
{{qed}}
[[Category:Modified Fort Spaces]]
[[Category:Examples of T0 Spaces]]
jcwy2h4yjjkh5rx6s4zac16xosmdwgw | Modified Fort Space is T0 | https://proofwiki.org/wiki/Modified_Fort_Space_is_T0 | https://proofwiki.org/wiki/Modified_Fort_Space_is_T0 | [
"Modified Fort Spaces",
"Examples of T0 Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:T0 Space"
] | [
"Modified Fort Space is T1",
"T1 Space is T0",
"Definition:T0 Space",
"Category:Modified Fort Spaces",
"Category:Examples of T0 Spaces"
] |
proofwiki-23455 | Modified Fort Space is not T3 | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is not a $T_3$ space. | We have that a Modified Fort Space is $T_0$.
We then have that a Modified Fort Space is not Regular.
{{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
But $T$ is a $T_0$ space and not a regular space.
So it follows that $T$ cannot be a $T_3$ space.
{{qed}} | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is not a [[Definition:T3 Space|$T_3$ space]]. | We have that a [[Modified Fort Space is T0|Modified Fort Space is $T_0$]].
We then have that a [[Modified Fort Space is not Regular]].
{{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
But $T$ is a [[Definition:T0 Space|$T_0$ space]] and not a [[Definition:Regular Space|regular space]].
... | Modified Fort Space is not T3 | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_T3 | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_T3 | [
"Modified Fort Spaces",
"Examples of T3 Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:T3 Space"
] | [
"Modified Fort Space is T0",
"Modified Fort Space is not Regular",
"Definition:T0 Space",
"Definition:Regular Space",
"Definition:T3 Space"
] |
proofwiki-23456 | Modified Fort Space is not Normal | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is not a normal space. | We have:
:Normal Space is $T_3$
:Modified Fort Space is not $T_3$
Hence the result.
{{qed}}
Category:Modified Fort Spaces
Category:Examples of Normal Spaces
oqyop9s6dzcd72lyybtd9ucjijp6iu5 | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is not a [[Definition:Normal Space|normal space]]. | We have:
:[[Normal Space is T3|Normal Space is $T_3$]]
:[[Modified Fort Space is not T3|Modified Fort Space is not $T_3$]]
Hence the result.
{{qed}}
[[Category:Modified Fort Spaces]]
[[Category:Examples of Normal Spaces]]
oqyop9s6dzcd72lyybtd9ucjijp6iu5 | Modified Fort Space is not Normal | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_Normal | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_Normal | [
"Modified Fort Spaces",
"Examples of Normal Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:Normal Space"
] | [
"Normal Space is T3",
"Modified Fort Space is not T3",
"Category:Modified Fort Spaces",
"Category:Examples of Normal Spaces"
] |
proofwiki-23457 | Modified Fort Space is not Regular | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is not a regular space. | {{Recall|Regular Space|regular space}}
{{:Definition:Regular Space/Definition 3}}
We have:
:Modified Fort Space is not $T_2$
Hence the result.
{{qed}}
Category:Modified Fort Spaces
Category:Examples of Regular Spaces
523jhrx0kkp85kunag14en0efc8m1yb | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is not a [[Definition:Regular Space|regular space]]. | {{Recall|Regular Space|regular space}}
{{:Definition:Regular Space/Definition 3}}
We have:
:[[Modified Fort Space is not T2|Modified Fort Space is not $T_2$]]
Hence the result.
{{qed}}
[[Category:Modified Fort Spaces]]
[[Category:Examples of Regular Spaces]]
523jhrx0kkp85kunag14en0efc8m1yb | Modified Fort Space is not Regular | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_Regular | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_Regular | [
"Modified Fort Spaces",
"Examples of Regular Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:Regular Space"
] | [
"Modified Fort Space is not T2",
"Category:Modified Fort Spaces",
"Category:Examples of Regular Spaces"
] |
proofwiki-23458 | Modified Fort Space is not T4 | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is not a $T_4$ space. | We have:
:Modified Fort Space is not Normal
:Modified Fort Space is $T_1$
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
But $T$ is a $T_1$ space which is not a normal space.
So it follows that $T$ cannot be a $T_4$ space.
{{qed}} | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is not a [[Definition:T4 Space|$T_4$ space]]. | We have:
:[[Modified Fort Space is not Normal]]
:[[Modified Fort Space is T1|Modified Fort Space is $T_1$]]
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
But $T$ is a [[Definition:T1 Space|$T_1$ space]] which is not a [[Definition:Normal Space|normal space]].
So it follows... | Modified Fort Space is not T4 | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_T4 | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_T4 | [
"Modified Fort Spaces",
"Examples of T4 Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:T4 Space"
] | [
"Modified Fort Space is not Normal",
"Modified Fort Space is T1",
"Definition:T1 Space",
"Definition:Normal Space",
"Definition:T4 Space"
] |
proofwiki-23459 | Modified Fort Space is not T5 | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is not a $T_5$ space. | We have:
:Modified Fort Space is not $T_4$
:$T_5$ Space is $T_4$
Hence the result.
{{qed}} | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is not a [[Definition:T5 Space|$T_5$ space]]. | We have:
:[[Modified Fort Space is not T4|Modified Fort Space is not $T_4$]]
:[[T5 Space is T4|$T_5$ Space is $T_4$]]
Hence the result.
{{qed}} | Modified Fort Space is not T5 | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_T5 | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_T5 | [
"Modified Fort Spaces",
"Examples of T5 Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:T5 Space"
] | [
"Modified Fort Space is not T4",
"T5 Space is T4"
] |
proofwiki-23460 | Modified Fort Space is not T2.5 | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is not a $T_{2 \frac 1 2}$ space. | We have:
:Modified Fort Space is not $T_2$
:$T_{2 \frac 1 2}$ Space is $T_2$
Hence the result.
{{qed}}
Category:Modified Fort Spaces
Category:Examples of T2.5 Spaces
jjc0pl9sbihif40wgfx3djy50x19jeb | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is not a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]]. | We have:
:[[Modified Fort Space is not T2|Modified Fort Space is not $T_2$]]
:[[T2.5 Space is T2|$T_{2 \frac 1 2}$ Space is $T_2$]]
Hence the result.
{{qed}}
[[Category:Modified Fort Spaces]]
[[Category:Examples of T2.5 Spaces]]
jjc0pl9sbihif40wgfx3djy50x19jeb | Modified Fort Space is not T2.5 | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_T2.5 | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_T2.5 | [
"Modified Fort Spaces",
"Examples of T2.5 Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:T2.5 Space"
] | [
"Modified Fort Space is not T2",
"T2.5 Space is T2",
"Category:Modified Fort Spaces",
"Category:Examples of T2.5 Spaces"
] |
proofwiki-23461 | Discrete Space is Perfectly Normal | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a perfectly normal space. | {{Recall|Perfectly Normal Space|perfectly normal space|index = 1}}
{{:Definition:Perfectly Normal Space/Definition 1}}
From Discrete Space is Perfectly $T_4$:
:$T$ is a perfectly $T_4$ space.
From Discrete Space is $T_1$:
:$T$ is a $T_1$ space.
The result follows.
{{qed}}
Category:Discrete Space satisfies all Separatio... | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:Perfectly Normal Space|perfectly normal space]]. | {{Recall|Perfectly Normal Space|perfectly normal space|index = 1}}
{{:Definition:Perfectly Normal Space/Definition 1}}
From [[Discrete Space is Perfectly T4|Discrete Space is Perfectly $T_4$]]:
:$T$ is a [[Definition:Perfectly T4 Space|perfectly $T_4$ space]].
From [[Discrete Space is T1|Discrete Space is $T_1$]]:
:$... | Discrete Space is Perfectly Normal | https://proofwiki.org/wiki/Discrete_Space_is_Perfectly_Normal | https://proofwiki.org/wiki/Discrete_Space_is_Perfectly_Normal | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of Perfectly Normal Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Perfectly Normal Space"
] | [
"Discrete Space is Perfectly T4",
"Definition:Perfectly T4 Space",
"Discrete Space is T1",
"Definition:T1 Space",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of Perfectly Normal Spaces"
] |
proofwiki-23462 | Discrete Space is Perfectly T4 | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a perfectly $T_4$ space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is Perfectly $T_4$.
{{qed}}
Category:Discrete Space satisfies all Separation Properties
Category:Discrete Topologies
Category:Examples of Perfectly T4 Spaces
7fu0d78os74btwrgj1hwdt94zq47532 | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:Perfectly T4 Space|perfectly $T_4$ space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is Perfectly T4|Metric Space is Perfectly $T_4$]].
{{qed}}
[[Category:Discrete Space satisfies all Separation Properties]]
[[Category:Discrete Topologies]]
[[Category:Examples of Perfectly T4 Spaces]]
7fu0d78... | Discrete Space is Perfectly T4 | https://proofwiki.org/wiki/Discrete_Space_is_Perfectly_T4 | https://proofwiki.org/wiki/Discrete_Space_is_Perfectly_T4 | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of Perfectly T4 Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Perfectly T4 Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is Perfectly T4",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of Perfectly T4 Spaces"
] |
proofwiki-23463 | Discrete Space is T0 | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a $T_0$ space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is $T_0$.
{{qed}}
Category:Discrete Space satisfies all Separation Properties
Category:Discrete Topologies
Category:Examples of T0 Spaces
14lps3o67ylt0umgz3afxs5f5j223mm | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is T0|Metric Space is $T_0$]].
{{qed}}
[[Category:Discrete Space satisfies all Separation Properties]]
[[Category:Discrete Topologies]]
[[Category:Examples of T0 Spaces]]
14lps3o67ylt0umgz3afxs5f5j223mm | Discrete Space is T0 | https://proofwiki.org/wiki/Discrete_Space_is_T0 | https://proofwiki.org/wiki/Discrete_Space_is_T0 | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of T0 Spaces"
] | [
"Definition:Discrete Topology",
"Definition:T0 Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is T0",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of T0 Spaces"
] |
proofwiki-23464 | Discrete Space is T1 | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a $T_1$ space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is $T_1$.
{{qed}} | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:T1 Space|$T_1$ space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is T1|Metric Space is $T_1$]].
{{qed}} | Discrete Space is T1 | https://proofwiki.org/wiki/Discrete_Space_is_T1 | https://proofwiki.org/wiki/Discrete_Space_is_T1 | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of T1 Spaces"
] | [
"Definition:Discrete Topology",
"Definition:T1 Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is T1"
] |
proofwiki-23465 | Discrete Space is T2 | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a $T_2$ (Hausdorff) space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is $T_2$.
{{qed}}
Category:Discrete Space satisfies all Separation Properties
Category:Discrete Topologies
Category:Examples of Hausdorff Spaces
jiee21lnl9kzptjqb1mbehkxt4jokmp | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is T2|Metric Space is $T_2$]].
{{qed}}
[[Category:Discrete Space satisfies all Separation Properties]]
[[Category:Discrete Topologies]]
[[Category:Examples of Hausdorff Spaces]]
jiee21lnl9kzptjqb1mbehkxt4jokm... | Discrete Space is T2 | https://proofwiki.org/wiki/Discrete_Space_is_T2 | https://proofwiki.org/wiki/Discrete_Space_is_T2 | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of Hausdorff Spaces"
] | [
"Definition:Discrete Topology",
"Definition:T2 Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is T2",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of Hausdorff Spaces"
] |
proofwiki-23466 | Discrete Space is Semiregular | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a semiregular space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is Semiregular.
{{qed}}
Category:Discrete Space satisfies all Separation Properties
Category:Discrete Topologies
Category:Examples of Semiregular Spaces
tihq69w1fk6xl4xxroxjpqmubr05f3q | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:Semiregular Space|semiregular space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is Semiregular]].
{{qed}}
[[Category:Discrete Space satisfies all Separation Properties]]
[[Category:Discrete Topologies]]
[[Category:Examples of Semiregular Spaces]]
tihq69w1fk6xl4xxroxjpqmubr05f3q | Discrete Space is Semiregular | https://proofwiki.org/wiki/Discrete_Space_is_Semiregular | https://proofwiki.org/wiki/Discrete_Space_is_Semiregular | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of Semiregular Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Semiregular Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is Semiregular",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of Semiregular Spaces"
] |
proofwiki-23467 | Discrete Space is T2.5 | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a $T_{2 \frac 1 2}$ space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is $T_{2 \frac 1 2}$.
{{qed}}
Category:Discrete Space satisfies all Separation Properties
Category:Discrete Topologies
Category:Examples of T2.5 Spaces
it6qzgs6rsmrvde7adi0cekobkui12b | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is T2.5|Metric Space is $T_{2 \frac 1 2}$]].
{{qed}}
[[Category:Discrete Space satisfies all Separation Properties]]
[[Category:Discrete Topologies]]
[[Category:Examples of T2.5 Spaces]]
it6qzgs6rsmrvde7adi0c... | Discrete Space is T2.5 | https://proofwiki.org/wiki/Discrete_Space_is_T2.5 | https://proofwiki.org/wiki/Discrete_Space_is_T2.5 | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of T2.5 Spaces"
] | [
"Definition:Discrete Topology",
"Definition:T2.5 Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is T2.5",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of T2.5 Spaces"
] |
proofwiki-23468 | Discrete Space is T3.5 | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a $T_{3 \frac 1 2}$ space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is $T_{3 \frac 1 2}$.
{{qed}}
Category:Discrete Space satisfies all Separation Properties
Category:Discrete Topologies
Category:Examples of T3.5 Spaces
0vt4ev7pelth3quz5vs2zoxvxikn2z5 | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is T3.5|Metric Space is $T_{3 \frac 1 2}$]].
{{qed}}
[[Category:Discrete Space satisfies all Separation Properties]]
[[Category:Discrete Topologies]]
[[Category:Examples of T3.5 Spaces]]
0vt4ev7pelth3quz5vs2z... | Discrete Space is T3.5 | https://proofwiki.org/wiki/Discrete_Space_is_T3.5 | https://proofwiki.org/wiki/Discrete_Space_is_T3.5 | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of T3.5 Spaces"
] | [
"Definition:Discrete Topology",
"Definition:T3.5 Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is T3.5",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of T3.5 Spaces"
] |
proofwiki-23469 | Discrete Space is T4 | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a $T_4$ space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is $T_4$ Space.
{{qed}}
Category:Discrete Space satisfies all Separation Properties
Category:Discrete Topologies
Category:Examples of T4 Spaces
logo1ub1kwnajkbpqr88dxbitzu320p | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:T4 Space|$T_4$ space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is T4 Space|Metric Space is $T_4$ Space]].
{{qed}}
[[Category:Discrete Space satisfies all Separation Properties]]
[[Category:Discrete Topologies]]
[[Category:Examples of T4 Spaces]]
logo1ub1kwnajkbpqr88dxbit... | Discrete Space is T4 | https://proofwiki.org/wiki/Discrete_Space_is_T4 | https://proofwiki.org/wiki/Discrete_Space_is_T4 | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of T4 Spaces"
] | [
"Definition:Discrete Topology",
"Definition:T4 Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is T4",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of T4 Spaces"
] |
proofwiki-23470 | Discrete Space is T5 | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a $T_5$ space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is $T_5$.
{{qed}}
Category:Discrete Space satisfies all Separation Properties
Category:Discrete Topologies
Category:Examples of T5 Spaces
l3e4gsx8vrfl9e89wdjwty10vzoesqp | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:T5 Space|$T_5$ space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is T5|Metric Space is $T_5$]].
{{qed}}
[[Category:Discrete Space satisfies all Separation Properties]]
[[Category:Discrete Topologies]]
[[Category:Examples of T5 Spaces]]
l3e4gsx8vrfl9e89wdjwty10vzoesqp | Discrete Space is T5 | https://proofwiki.org/wiki/Discrete_Space_is_T5 | https://proofwiki.org/wiki/Discrete_Space_is_T5 | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of T5 Spaces"
] | [
"Definition:Discrete Topology",
"Definition:T5 Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is T5",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of T5 Spaces"
] |
proofwiki-23471 | Discrete Space is Regular | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a regular space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is Regular Space.
{{qed}}
Category:Discrete Space satisfies all Separation Properties
Category:Discrete Topologies
Category:Examples of Regular Spaces
tvmv0jwemimffmkira0r5kyqifub5hb | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:Regular Space|regular space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is Regular Space]].
{{qed}}
[[Category:Discrete Space satisfies all Separation Properties]]
[[Category:Discrete Topologies]]
[[Category:Examples of Regular Spaces]]
tvmv0jwemimffmkira0r5kyqifub5hb | Discrete Space is Regular | https://proofwiki.org/wiki/Discrete_Space_is_Regular | https://proofwiki.org/wiki/Discrete_Space_is_Regular | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of Regular Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Regular Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is Regular",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of Regular Spaces"
] |
proofwiki-23472 | Discrete Space is Urysohn | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is an Urysohn space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is Urysohn.
{{qed}}
Category:Discrete Space satisfies all Separation Properties
Category:Discrete Topologies
Category:Examples of Urysohn Spaces
3jtw0ar6aduz0skw5pt0ntkiysxtxkc | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is an [[Definition:Urysohn Space|Urysohn space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is Urysohn]].
{{qed}}
[[Category:Discrete Space satisfies all Separation Properties]]
[[Category:Discrete Topologies]]
[[Category:Examples of Urysohn Spaces]]
3jtw0ar6aduz0skw5pt0ntkiysxtxkc | Discrete Space is Urysohn | https://proofwiki.org/wiki/Discrete_Space_is_Urysohn | https://proofwiki.org/wiki/Discrete_Space_is_Urysohn | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of Urysohn Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Urysohn Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is Urysohn",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of Urysohn Spaces"
] |
proofwiki-23473 | Discrete Space is Completely Regular | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a completely regular space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is Completely Regular.
{{qed}}
Category:Discrete Space satisfies all Separation Properties
Category:Discrete Topologies
Category:Examples of Completely Regular Spaces
32yy8u8qtlpi8ywibyhm99r1c21sonl | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:Completely Regular Space|completely regular space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is Completely Regular]].
{{qed}}
[[Category:Discrete Space satisfies all Separation Properties]]
[[Category:Discrete Topologies]]
[[Category:Examples of Completely Regular Spaces]]
32yy8u8qtlpi8ywibyhm99r1c21... | Discrete Space is Completely Regular | https://proofwiki.org/wiki/Discrete_Space_is_Completely_Regular | https://proofwiki.org/wiki/Discrete_Space_is_Completely_Regular | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of Completely Regular Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Completely Regular Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is Completely Regular",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of Completely Regular Spaces"
] |
proofwiki-23474 | Discrete Space is Completely Normal | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a completely normal space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is Completely Normal.
{{qed}}
Category:Discrete Space satisfies all Separation Properties
Category:Discrete Topologies
Category:Examples of Completely Normal Spaces
qarjdg84medbnfxk9hetz0nirigy5md | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:Completely Normal Space|completely normal space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is Completely Normal]].
{{qed}}
[[Category:Discrete Space satisfies all Separation Properties]]
[[Category:Discrete Topologies]]
[[Category:Examples of Completely Normal Spaces]]
qarjdg84medbnfxk9hetz0nirigy5... | Discrete Space is Completely Normal | https://proofwiki.org/wiki/Discrete_Space_is_Completely_Normal | https://proofwiki.org/wiki/Discrete_Space_is_Completely_Normal | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of Completely Normal Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Completely Normal Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is Completely Normal",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of Completely Normal Spaces"
] |
proofwiki-23475 | Discrete Space is Normal | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ is a normal space. | We have that a Standard Discrete Metric induces Discrete Topology.
The result follows from Metric Space is Normal Space.
{{qed}}
Category:Discrete Space satisfies all Separation Properties
Category:Discrete Topologies
Category:Examples of Normal Spaces
gf5e76vtgowo0p4ozgr9c8y4ritnoce | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ is a [[Definition:Normal Space|normal space]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
The result follows from [[Metric Space is Normal Space]].
{{qed}}
[[Category:Discrete Space satisfies all Separation Properties]]
[[Category:Discrete Topologies]]
[[Category:Examples of Normal Spaces]]
gf5e76vtgowo0p4ozgr9c8y4ritnoce | Discrete Space is Normal | https://proofwiki.org/wiki/Discrete_Space_is_Normal | https://proofwiki.org/wiki/Discrete_Space_is_Normal | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of Normal Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Normal Space"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space is Normal",
"Category:Discrete Space satisfies all Separation Properties",
"Category:Discrete Topologies",
"Category:Examples of Normal Spaces"
] |
proofwiki-23476 | Reflection Formula (Geometry) | Let $H$ be a Hilbert space.
Let $v, a, \in H$ be arbitrary elements of $H$.
Let $A \subset H$ denote the hyperplane orthogonal to $a$.
The reflection of $v$ through $A$ is given by:
:$\map {\operatorname {Ref}_a} v = v - 2 \dfrac {\langle v, a \rangle} {\langle a, a \rangle} a$ | By Orthogonal Projection onto Orthocomplement, we may write:
:$v = v^\parallel + v^\perp$
such that $v^\perp \in A$ and $v^\parallel \in A^\perp$.
$v^\parallel$ is the projection of $v$ onto $a$ given by:
:$v^\parallel = \dfrac {\langle v, a\rangle} {\langle a, a \rangle} a$
{{explain|What is $v^\perp$?}}
Then:
:$\map ... | Let $H$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $v, a, \in H$ be [[Definition:Arbitrary|arbitrary]] [[Definition:Element|elements]] of $H$.
Let $A \subset H$ denote the [[Definition:Hyperplane|hyperplane]] [[Definition:Orthogonal|orthogonal]] to $a$.
The [[Definition:Reflection (Geometry)|reflection]] ... | By [[Orthogonal Projection onto Orthocomplement]], we may write:
:$v = v^\parallel + v^\perp$
such that $v^\perp \in A$ and $v^\parallel \in A^\perp$.
$v^\parallel$ is the [[Definition:Vector Projection/Definition 2|projection]] of $v$ onto $a$ given by:
:$v^\parallel = \dfrac {\langle v, a\rangle} {\langle a, a \rang... | Reflection Formula (Geometry) | https://proofwiki.org/wiki/Reflection_Formula_(Geometry) | https://proofwiki.org/wiki/Reflection_Formula_(Geometry) | [
"Orthogonal Projections",
"Orthocomplements",
"Geometric Reflections",
"Isometries (Euclidean Geometry)",
"Euclidean Geometry"
] | [
"Definition:Hilbert Space",
"Definition:Arbitrary",
"Definition:Element",
"Definition:Hyperplane",
"Definition:Orthogonal",
"Definition:Reflection (Geometry)"
] | [
"Orthogonal Projection onto Orthocomplement",
"Definition:Vector Projection/Definition 2",
"Category:Orthogonal Projections",
"Category:Orthocomplements",
"Category:Geometric Reflections",
"Category:Isometries (Euclidean Geometry)",
"Category:Euclidean Geometry"
] |
proofwiki-23477 | Cartan Subalgebra as Maximal Abelian Subalgebra | Let $\mathfrak g$ be a finite-dimensional, semisimple Lie algebra over an algebraically closed field.
Let $\mathfrak h \subset \mathfrak g$ be a Lie subalgebra.
$\mathfrak h$ is a Cartan subalgebra of $\mathfrak{g}$ {{iff}}:
{{begin-axiom}}
{{axiom | n = 1 | t = $\mathfrak h$ is a maximal Abelian subalgebra of $\mathfr... | {{ProofWanted}}
Category:Cartan Subalgebras
Category:Lie Subalgebras
ftb08i9k1jptgxmlc56s5kje7zkedim | Let $\mathfrak g$ be a [[Definition:Finite Dimensional Vector Space|finite-dimensional]], [[Definition:Semisimple Lie Algebra|semisimple]] [[Definition:Lie Algebra|Lie algebra]] over an [[Definition:Algebraically Closed Field|algebraically closed field]].
Let $\mathfrak h \subset \mathfrak g$ be a [[Definition:Lie Sub... | {{ProofWanted}}
[[Category:Cartan Subalgebras]]
[[Category:Lie Subalgebras]]
ftb08i9k1jptgxmlc56s5kje7zkedim | Cartan Subalgebra as Maximal Abelian Subalgebra | https://proofwiki.org/wiki/Cartan_Subalgebra_as_Maximal_Abelian_Subalgebra | https://proofwiki.org/wiki/Cartan_Subalgebra_as_Maximal_Abelian_Subalgebra | [
"Cartan Subalgebras",
"Lie Subalgebras"
] | [
"Definition:Dimension of Vector Space/Finite",
"Definition:Semisimple Lie Algebra",
"Definition:Lie Algebra",
"Definition:Algebraically Closed Field",
"Definition:Lie Subalgebra",
"Definition:Cartan Subalgebra",
"Definition:Maximal Abelian Subalgebra",
"Definition:Adjoint Representation (Lie Algebra)"... | [
"Category:Cartan Subalgebras",
"Category:Lie Subalgebras"
] |
proofwiki-23478 | Weyl Group Permutes Root System | Let $\struct {E, \Phi}$ be a root system with Weyl group $\mathscr W$.
Then $\mathscr W$ permutes $\Phi$. | {{tidy}}
Let $\sigma$ be an element of $\mathscr W$.
$\map \sigma \Phi \subset \Phi$ by Axiom (3) of a Root System.
Since $\sigma \in \mathscr W \subset \map {GL} E$, its elements are invertible by definition of the general linear group over $E$.
The result follows by Inverse Mapping is Bijection.
{{qed}}
Category:Weyl... | Let $\struct {E, \Phi}$ be a [[Definition:Root System|root system]] with [[Definition:Weyl Group|Weyl group]] $\mathscr W$.
Then $\mathscr W$ [[Definition:Permutation|permutes]] $\Phi$. | {{tidy}}
Let $\sigma$ be an [[Definition:Element|element]] of $\mathscr W$.
$\map \sigma \Phi \subset \Phi$ by [[Definition:Root System|Axiom (3) of a Root System]].
Since $\sigma \in \mathscr W \subset \map {GL} E$, its [[Definition:Element|elements]] are [[Definition:Inverse Mapping|invertible]] by definition of t... | Weyl Group Permutes Root System | https://proofwiki.org/wiki/Weyl_Group_Permutes_Root_System | https://proofwiki.org/wiki/Weyl_Group_Permutes_Root_System | [
"Weyl Groups",
"Root Systems"
] | [
"Definition:Root System",
"Definition:Weyl Group",
"Definition:Permutation"
] | [
"Definition:Element",
"Definition:Root System",
"Definition:Element",
"Definition:Inverse Mapping",
"Definition:General Linear Group/Vector Space",
"Inverse Mapping is Bijection",
"Category:Weyl Groups",
"Category:Root Systems"
] |
proofwiki-23479 | Root System has Base | Let $\struct {E, \Phi}$ be a root system with base $\Delta \subset \Phi$.
Let $P_\alpha$ denote the hyperplane orthogonal to $\alpha\in\Phi$.
{{Disambiguate|Definition:Orthogonal}}
For $\gamma \in E$, let $\map {\Phi^+} \gamma : = \set {\alpha \in \Phi: \innerprod \alpha \gamma > 0}$.
Call $\gamma$ '''regular''' {{iff}... | By Vector Space over an Infinite Field is not equal to the Union of Proper Subspaces, regular elements of $E$ exist.
{{ProofWanted}} | Let $\struct {E, \Phi}$ be a [[Definition:Root System|root system]] with [[Definition:Base of Root System|base]] $\Delta \subset \Phi$.
Let $P_\alpha$ denote the [[Definition:Hyperplane|hyperplane]] [[Definition:Orthogonal|orthogonal]] to $\alpha\in\Phi$.
{{Disambiguate|Definition:Orthogonal}}
For $\gamma \in E$, le... | By [[Vector Space over an Infinite Field is not equal to the Union of Proper Subspaces]], regular elements of $E$ exist.
{{ProofWanted}} | Root System has Base | https://proofwiki.org/wiki/Root_System_has_Base | https://proofwiki.org/wiki/Root_System_has_Base | [
"Root Systems"
] | [
"Definition:Root System",
"Definition:Base of Root System",
"Definition:Hyperplane",
"Definition:Orthogonal",
"Definition:Base of Root System",
"Definition:Base of Root System"
] | [
"Vector Space over an Infinite Field is not equal to the Union of Proper Subspaces"
] |
proofwiki-23480 | Complete Metric Space is Non-Meager | Let $M = \struct {A, d}$ be a complete metric space.
Then $M$ is non-meager. | From the Baire Category Theorem for Complete Metric Spaces, $\struct {S, d}$ is a Baire space.
By Baire Space is Non-Meager, it follows that $T$ is non-meager.
{{qed}}
{{ADC|Baire Space is Non-Meager}} | Let $M = \struct {A, d}$ be a [[Definition:Complete Metric Space|complete metric space]].
Then $M$ is [[Definition:Non-Meager Space|non-meager]]. | From the [[Baire Category Theorem for Complete Metric Spaces]], $\struct {S, d}$ is a [[Definition:Baire Space (Topology)|Baire space]].
By [[Baire Space is Non-Meager]], it follows that $T$ is [[Definition:Non-Meager Space|non-meager]].
{{qed}}
{{ADC|Baire Space is Non-Meager}} | Complete Metric Space is Non-Meager | https://proofwiki.org/wiki/Complete_Metric_Space_is_Non-Meager | https://proofwiki.org/wiki/Complete_Metric_Space_is_Non-Meager | [
"Complete Metric Spaces",
"Non-Meager Spaces",
"Examples of Use of Axiom of Dependent Choice"
] | [
"Definition:Complete Metric Space",
"Definition:Meager Space/Non-Meager"
] | [
"Baire Category Theorem/Complete Metric Space",
"Definition:Baire Space (Topology)",
"Baire Space is Non-Meager",
"Definition:Meager Space/Non-Meager"
] |
proofwiki-23481 | Hilbert Cube is Locally Injectively Path-Connected | Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube.
Then $M$ is a locally injectively path-connected space. | {{Recall|Locally Injectively Path-Connected Space|locally injectively path-connected space}}
{{:Definition:Locally Injectively Path-Connected Space}}
Let $\alpha \in I^\omega$ be arbitrary.
Let $\map {B_\epsilon} \alpha$ be an open ball of $\alpha$ in $M$.
Let $y \in \map {B_\epsilon} \alpha$ be arbitrary.
We have:
:$\... | Let $M = \struct {I^\omega, d_2}$ be the [[Definition:Hilbert Cube|Hilbert cube]].
Then $M$ is a [[Definition:Locally Injectively Path-Connected Space|locally injectively path-connected space]]. | {{Recall|Locally Injectively Path-Connected Space|locally injectively path-connected space}}
{{:Definition:Locally Injectively Path-Connected Space}}
Let $\alpha \in I^\omega$ be [[Definition:Arbitrary|arbitrary]].
Let $\map {B_\epsilon} \alpha$ be an [[Definition:Open Ball|open ball]] of $\alpha$ in $M$.
Let $y \in... | Hilbert Cube is Locally Injectively Path-Connected | https://proofwiki.org/wiki/Hilbert_Cube_is_Locally_Injectively_Path-Connected | https://proofwiki.org/wiki/Hilbert_Cube_is_Locally_Injectively_Path-Connected | [
"Hilbert Cube",
"Examples of Locally Injectively Path-Connected Spaces"
] | [
"Definition:Hilbert Cube",
"Definition:Locally Injectively Path-Connected Space"
] | [
"Definition:Arbitrary",
"Definition:Open Ball",
"Definition:Arbitrary",
"Definition:Line/Straight Line Segment",
"Open Balls form Basis for Open Sets of Metric Space",
"Definition:Locally Injectively Path-Connected Space"
] |
proofwiki-23482 | Linearly Ordered Space is Regular | Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space.
Then $\struct {S, \tau}$ is regular. | From Linearly Ordered Space is Normal, $\struct {S, \tau}$ is a normal space.
The result follows from Normal Space is Regular.
{{qed}}
Category:Linearly Ordered Spaces
Category:Examples of Regular Spaces
eka6nca65nyxkdr4pb4bh8ig3qdecuz | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Linearly Ordered Space|linearly ordered space]].
Then $\struct {S, \tau}$ is [[Definition:Regular Space|regular]]. | From [[Linearly Ordered Space is Normal]], $\struct {S, \tau}$ is a [[Definition:Normal Space|normal space]].
The result follows from [[Normal Space is Regular]].
{{qed}}
[[Category:Linearly Ordered Spaces]]
[[Category:Examples of Regular Spaces]]
eka6nca65nyxkdr4pb4bh8ig3qdecuz | Linearly Ordered Space is Regular | https://proofwiki.org/wiki/Linearly_Ordered_Space_is_Regular | https://proofwiki.org/wiki/Linearly_Ordered_Space_is_Regular | [
"Linearly Ordered Spaces",
"Examples of Regular Spaces"
] | [
"Definition:Linearly Ordered Space",
"Definition:Regular Space"
] | [
"Linearly Ordered Space is Normal",
"Definition:Normal Space",
"Normal Space is Regular",
"Category:Linearly Ordered Spaces",
"Category:Examples of Regular Spaces"
] |
proofwiki-23483 | Ordinal Space is Regular | Let $\Gamma$ denote a limit ordinal.
Let $\hointr 0 \Gamma$ denote the open ordinal space on $\Gamma$.
Let $\closedint 0 \Gamma$ denote the closed ordinal space on $\Gamma$.
Then $\hointr 0 \Gamma$ and $\closedint 0 \Gamma$ are both regular spaces. | By definition, $\hointr 0 \Gamma$ and $\closedint 0 \Gamma$ are both linearly ordered spaces.
The result follows from Linearly Ordered Space is Regular.
{{qed}}
Category:Ordinal Spaces
Category:Examples of Regular Spaces
cyxxvsr3hccqkeepy2w93wym3tjmpnt | Let $\Gamma$ denote a [[Definition:Limit Ordinal|limit ordinal]].
Let $\hointr 0 \Gamma$ denote the [[Definition:Open Ordinal Space|open ordinal space]] on $\Gamma$.
Let $\closedint 0 \Gamma$ denote the [[Definition:Closed Ordinal Space|closed ordinal space]] on $\Gamma$.
Then $\hointr 0 \Gamma$ and $\closedint 0 \... | By definition, $\hointr 0 \Gamma$ and $\closedint 0 \Gamma$ are both [[Definition:Linearly Ordered Space|linearly ordered spaces]].
The result follows from [[Linearly Ordered Space is Regular]].
{{qed}}
[[Category:Ordinal Spaces]]
[[Category:Examples of Regular Spaces]]
cyxxvsr3hccqkeepy2w93wym3tjmpnt | Ordinal Space is Regular | https://proofwiki.org/wiki/Ordinal_Space_is_Regular | https://proofwiki.org/wiki/Ordinal_Space_is_Regular | [
"Ordinal Spaces",
"Examples of Regular Spaces"
] | [
"Definition:Limit Ordinal",
"Definition:Ordinal Space/Open",
"Definition:Ordinal Space/Closed",
"Definition:Regular Space"
] | [
"Definition:Linearly Ordered Space",
"Linearly Ordered Space is Regular",
"Category:Ordinal Spaces",
"Category:Examples of Regular Spaces"
] |
proofwiki-23484 | Set with no Limit Point is Closed | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be such that $H$ has no limit point.
Then $H$ is a closed set of $T$. | {{Recall|Closed Set (Topology)|closed set}}
{{:Definition:Closed Set (Topology)/Definition 2}}
But {{hypothesis}} $H$ has no limit point.
Hence vacuously $H$ contains all its limit points.
Hence the result.
{{qed}}
Category:Limit Points of Sets
Category:Closed Sets
n2ueqsosjp8s7sq3r6lp395vm1eqj0k | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$ be such that $H$ has no [[Definition:Limit Point of Set|limit point]].
Then $H$ is a [[Definition:Closed Set (Topology)|closed set]] of $T$. | {{Recall|Closed Set (Topology)|closed set}}
{{:Definition:Closed Set (Topology)/Definition 2}}
But {{hypothesis}} $H$ has no [[Definition:Limit Point of Set|limit point]].
Hence [[Definition:Vacuous Truth|vacuously]] $H$ contains all its [[Definition:Limit Point of Set|limit points]].
Hence the result.
{{qed}}
[[Ca... | Set with no Limit Point is Closed | https://proofwiki.org/wiki/Set_with_no_Limit_Point_is_Closed | https://proofwiki.org/wiki/Set_with_no_Limit_Point_is_Closed | [
"Limit Points of Sets",
"Closed Sets"
] | [
"Definition:Topological Space",
"Definition:Limit Point/Topology/Set",
"Definition:Closed Set/Topology"
] | [
"Definition:Limit Point/Topology/Set",
"Definition:Vacuous Truth",
"Definition:Limit Point/Topology/Set",
"Category:Limit Points of Sets",
"Category:Closed Sets"
] |
proofwiki-23485 | Equivalence of Definitions of T3.5 Space | {{TFAE|def = T3.5 Space|view = a $T_{3 \frac 1 2}$ space}}
Let $T = \struct {S, \tau}$ be a topological space. | === Definition 1 implies Definition 2 ===
Let $T = \struct {S, \tau}$ be a $T_{3 \frac 1 2}$ space by definition $1$.
{{Recall|T3.5 Space|$T_{3 \frac 1 2}$ space|index = 1}}
{{:Definition:T3.5 Space/Definition 1}}
{{ProofWanted}}
{{Recall|T3.5 Space|$T_{3 \frac 1 2}$ space|index = 2}}
{{:Definition:T3.5 Space/Definitio... | {{TFAE|def = T3.5 Space|view = a $T_{3 \frac 1 2}$ space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]]. | === Definition 1 implies Definition 2 ===
Let $T = \struct {S, \tau}$ be a [[Definition:T3.5 Space/Definition 1|$T_{3 \frac 1 2}$ space by definition $1$]].
{{Recall|T3.5 Space|$T_{3 \frac 1 2}$ space|index = 1}}
{{:Definition:T3.5 Space/Definition 1}}
{{ProofWanted}}
{{Recall|T3.5 Space|$T_{3 \frac 1 2}$ space|ind... | Equivalence of Definitions of T3.5 Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_T3.5_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_T3.5_Space | [
"T3.5 Spaces"
] | [
"Definition:Topological Space"
] | [
"Definition:T3.5 Space/Definition 1",
"Definition:T3.5 Space/Definition 2",
"Definition:T3.5 Space/Definition 2",
"Definition:T3.5 Space/Definition 1"
] |
proofwiki-23486 | Topological Space is Compact iff Every Ultrafilter Converges | A topological space $T = \struct {S, \tau}$ is '''compact''' {{iff}} every ultrafilter on $S$ converges to a point in $S$. | From Topological Space is Compact iff Every Filter has Adherent Point, every filter on $S$ has an adherent point in $S$.
It remains to be proved that:
:every filter on $S$ has an adherent point in $S$ {{iff}} every ultrafilter on $S$ converges to a point in $S$. | A [[Definition:Topological Space|topological space]] $T = \struct {S, \tau}$ is '''[[Definition:Compact Topological Space|compact]]''' {{iff}} every [[Definition:Ultrafilter on Set|ultrafilter]] on $S$ [[Definition:Convergent Filter|converges]] to a [[Definition:Point of Set|point]] in $S$. | From [[Topological Space is Compact iff Every Filter has Adherent Point]], every [[Definition:Filter on Set|filter]] on $S$ has an [[Definition:Adherent Point of Filter|adherent point]] in $S$.
It remains to be proved that:
:every [[Definition:Filter on Set|filter]] on $S$ has an [[Definition:Adherent Point of Filter... | Topological Space is Compact iff Every Ultrafilter Converges | https://proofwiki.org/wiki/Topological_Space_is_Compact_iff_Every_Ultrafilter_Converges | https://proofwiki.org/wiki/Topological_Space_is_Compact_iff_Every_Ultrafilter_Converges | [
"Topological Space is Compact iff Every Ultrafilter Converges",
"Compact Topological Spaces",
"Ultrafilters on Sets"
] | [
"Definition:Topological Space",
"Definition:Compact Topological Space",
"Definition:Ultrafilter on Set",
"Definition:Convergent Filter",
"Definition:Element"
] | [
"Topological Space is Compact iff Every Filter has Adherent Point",
"Definition:Filter on Set",
"Definition:Adherent Point/Filter",
"Definition:Filter on Set",
"Definition:Adherent Point/Filter",
"Definition:Ultrafilter on Set",
"Definition:Convergent Filter",
"Definition:Element",
"Definition:Ultra... |
proofwiki-23487 | Hermitian Matrix is Hermitian Operator | A Hermitian matrix is a Hermitian operator. | Let $\mathbf A \in \mathbb C^{n \times n}$ be a Hermitian matrix.
Let $\mathbf v, \mathbf w \in \mathbb C^n$ be vectors.
Then:
{{begin-eqn}}
{{eqn | l = \innerprod {\mathbf A \mathbf v} {\mathbf w}
| r = \mathbf w^\dagger \paren {\mathbf A \mathbf v}
| c = {{Defof|Complex Vector Inner Product}}
}}
{{eqn | r... | A [[Definition:Hermitian Matrix|Hermitian matrix]] is a [[Definition:Hermitian Operator|Hermitian operator]]. | Let $\mathbf A \in \mathbb C^{n \times n}$ be a [[Definition:Hermitian Matrix|Hermitian matrix]].
Let $\mathbf v, \mathbf w \in \mathbb C^n$ be [[Definition:Vector|vectors]].
Then:
{{begin-eqn}}
{{eqn | l = \innerprod {\mathbf A \mathbf v} {\mathbf w}
| r = \mathbf w^\dagger \paren {\mathbf A \mathbf v}
|... | Hermitian Matrix is Hermitian Operator | https://proofwiki.org/wiki/Hermitian_Matrix_is_Hermitian_Operator | https://proofwiki.org/wiki/Hermitian_Matrix_is_Hermitian_Operator | [
"Hermitian Operators",
"Hermitian Matrices",
"Complex Vector Spaces"
] | [
"Definition:Hermitian Matrix",
"Definition:Hermitian Operator"
] | [
"Definition:Hermitian Matrix",
"Definition:Vector",
"Matrix Multiplication is Associative",
"Hermitian Conjugate of Matrix Product",
"Hermitian Conjugate is Involution",
"Definition:Adjoint Linear Transformation",
"Definition:Hermitian Operator",
"Category:Hermitian Operators",
"Category:Hermitian M... |
proofwiki-23488 | Hermitian Conjugate of Matrix Product | Let $\mathbf A$ and $\mathbf B$ be matrices over a commutative ring such that $\mathbf A \mathbf B$ is defined.
Then $\mathbf B^\dagger \mathbf A^\dagger$ is defined, and:
:$\paren {\mathbf A \mathbf B}^\dagger = \mathbf B^\dagger \mathbf A^\dagger$
where $\mathbf X^\dagger$ is the Hermitian conjugate of $\mathbf X$. | We have:
{{begin-eqn}}
{{eqn | l = \paren {\mathbf A \mathbf B}^\dagger
| r = \overline {\paren {\mathbf A \mathbf B}^\intercal}
| c = {{Defof|Hermitian Conjugate}}
}}
{{eqn | r = \overline {\mathbf B^\intercal \mathbf A^\intercal}
| c = Transpose of Matrix Product
}}
{{eqn | r = \overline {\mathbf B}... | Let $\mathbf A$ and $\mathbf B$ be [[Definition:Matrix|matrices]] over a [[Definition:Commutative Ring|commutative ring]] such that $\mathbf A \mathbf B$ is [[Definition:Matrix Product (Conventional)|defined]].
Then $\mathbf B^\dagger \mathbf A^\dagger$ is [[Definition:Matrix Product (Conventional)|defined]], and:
:$... | We have:
{{begin-eqn}}
{{eqn | l = \paren {\mathbf A \mathbf B}^\dagger
| r = \overline {\paren {\mathbf A \mathbf B}^\intercal}
| c = {{Defof|Hermitian Conjugate}}
}}
{{eqn | r = \overline {\mathbf B^\intercal \mathbf A^\intercal}
| c = [[Transpose of Matrix Product]]
}}
{{eqn | r = \overline {\mathb... | Hermitian Conjugate of Matrix Product | https://proofwiki.org/wiki/Hermitian_Conjugate_of_Matrix_Product | https://proofwiki.org/wiki/Hermitian_Conjugate_of_Matrix_Product | [
"Hermitian Conjugates",
"Conventional Matrix Multiplication"
] | [
"Definition:Matrix",
"Definition:Commutative Ring",
"Definition:Matrix Product (Conventional)",
"Definition:Matrix Product (Conventional)",
"Definition:Hermitian Conjugate"
] | [
"Transpose of Matrix Product",
"Product of Complex Conjugates",
"Category:Hermitian Conjugates",
"Category:Conventional Matrix Multiplication"
] |
proofwiki-23489 | Arc-Connected Space is Path-Connected | Let $T = \struct {S, \tau}$ be a topological space which is arc-connected.
Then $T$ is path-connected. | Let $T = \struct {S, \tau}$ be arc-connected.
Then $\forall x, y \in S$, there exists an arc $f: \closedint 0 1 \to S$ such that $\map f 0 = x$ and $\map f 1 = y$.
From Arc is Injective Path, $f$ is {{afortiori}} an injective path.
The result follows from Injectively Path-Connected Space is Path-Connected.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Arc-Connected Space|arc-connected]].
Then $T$ is [[Definition:Path-Connected Space|path-connected]]. | Let $T = \struct {S, \tau}$ be [[Definition:Arc-Connected Space|arc-connected]].
Then $\forall x, y \in S$, there exists an [[Definition:Arc (Topology)|arc]] $f: \closedint 0 1 \to S$ such that $\map f 0 = x$ and $\map f 1 = y$.
From [[Arc is Injective Path]], $f$ is {{afortiori}} an [[Definition:Injective Path|injec... | Arc-Connected Space is Path-Connected | https://proofwiki.org/wiki/Arc-Connected_Space_is_Path-Connected | https://proofwiki.org/wiki/Arc-Connected_Space_is_Path-Connected | [
"Arc-Connected Spaces",
"Path-Connected Spaces",
"Sequence of Implications of Connectedness Properties"
] | [
"Definition:Topological Space",
"Definition:Arc-Connected/Topological Space",
"Definition:Path-Connected/Topological Space"
] | [
"Definition:Arc-Connected/Topological Space",
"Definition:Arc (Topology)",
"Arc is Injective Path",
"Definition:Injective Path",
"Injectively Path-Connected Space is Path-Connected"
] |
proofwiki-23490 | Joining Arcs makes Another Arc | Let $T$ be a topological space.
Let $\mathbb I \subseteq \R$ be the closed unit interval $\closedint 0 1$.
Let $a, b, c$ be three distinct points of $T$.
Let $f, g: \mathbb I \to T$ be arcs in $T$ from $a$ to $b$ and from $b$ to $c$ respectively.
Let $h: \mathbb I \to T$ be the mapping given by:
:<nowiki>$\map h x = \b... | {{ProofWanted}}
Category:Arcs (Topology)
n5bxcpqirj8m3kvlv7vuib2tptlvmqe | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $\mathbb I \subseteq \R$ be the [[Definition:Closed Unit Interval|closed unit interval]] $\closedint 0 1$.
Let $a, b, c$ be three distinct points of $T$.
Let $f, g: \mathbb I \to T$ be [[Definition:Arc (Topology)|arcs]] in $T$ from $a$ to $b$ and f... | {{ProofWanted}}
[[Category:Arcs (Topology)]]
n5bxcpqirj8m3kvlv7vuib2tptlvmqe | Joining Arcs makes Another Arc | https://proofwiki.org/wiki/Joining_Arcs_makes_Another_Arc | https://proofwiki.org/wiki/Joining_Arcs_makes_Another_Arc | [
"Arcs (Topology)"
] | [
"Definition:Topological Space",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Arc (Topology)",
"Definition:Mapping",
"Definition:Arc (Topology)",
"Definition:Restriction/Mapping",
"Definition:Arc (Topology)"
] | [
"Category:Arcs (Topology)"
] |
proofwiki-23491 | Arc-Connectedness is Equivalence Relation | Let $T = \struct {S, \tau}$ be a topological space.
Let $a \sim b $ denote the relation:
:$a \sim b \iff a$ is arc-connected to $b$
where $a, b \in S$.
Then $\sim$ is an equivalence relation. | {{ProofWanted|While it is expected that this is similar to Injective Path-Connectedness is Equivalence Relation, no assumptions made until it has been thought about}}
Category:Examples of Equivalence Relations
Category:Arc-Connectedness
gs3kxsjw86hh21zqpy8iqc8h8sd3ji7 | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $a \sim b $ denote the [[Definition:Relation|relation]]:
:$a \sim b \iff a$ is [[Definition:Arc-Connected Points|arc-connected]] to $b$
where $a, b \in S$.
Then $\sim$ is an [[Definition:Equivalence Relation|equivalence relation... | {{ProofWanted|While it is expected that this is similar to [[Injective Path-Connectedness is Equivalence Relation]], no assumptions made until it has been thought about}}
[[Category:Examples of Equivalence Relations]]
[[Category:Arc-Connectedness]]
gs3kxsjw86hh21zqpy8iqc8h8sd3ji7 | Arc-Connectedness is Equivalence Relation | https://proofwiki.org/wiki/Arc-Connectedness_is_Equivalence_Relation | https://proofwiki.org/wiki/Arc-Connectedness_is_Equivalence_Relation | [
"Examples of Equivalence Relations",
"Arc-Connectedness"
] | [
"Definition:Topological Space",
"Definition:Relation",
"Definition:Arc-Connected/Points",
"Definition:Equivalence Relation"
] | [
"Injective Path-Connectedness is Equivalence Relation",
"Category:Examples of Equivalence Relations",
"Category:Arc-Connectedness"
] |
proofwiki-23492 | Complex Vector Inner Product is Complex Inner Product | The complex vector inner product is an example of a complex inner product. | {{Recall|Complex Vector Inner Product|complex vector inner product}}
{{:Definition:Complex Vector Inner Product}}
Let $\mathbf x, \mathbf y, \mathbf z \in \mathbb C^n$ be vectors. | The [[Definition:Complex Vector Inner Product|complex vector inner product]] is an example of a [[Definition:Complex Inner Product|complex inner product]]. | {{Recall|Complex Vector Inner Product|complex vector inner product}}
{{:Definition:Complex Vector Inner Product}}
Let $\mathbf x, \mathbf y, \mathbf z \in \mathbb C^n$ be [[Definition:Vector|vectors]]. | Complex Vector Inner Product is Complex Inner Product | https://proofwiki.org/wiki/Complex_Vector_Inner_Product_is_Complex_Inner_Product | https://proofwiki.org/wiki/Complex_Vector_Inner_Product_is_Complex_Inner_Product | [
"Complex Vector Inner Product",
"Inner Products",
"Complex Vector Spaces"
] | [
"Definition:Inner Product/Complex Vector Space",
"Definition:Inner Product/Complex Field"
] | [
"Definition:Vector",
"Definition:Inner Product/Complex Vector Space"
] |
proofwiki-23493 | Number of Roots of Polynomial With Complex Coefficients Equals Degree of Polynomial | Every non-constant polynomial of degree $n > 0$ with coefficients in $\C$ has $n$ roots in $\C$ including multiple roots. | Proof by induction:
For all $n \in \mathbb N_{>0}$, let $\map P n$ be the proposition:
:$\map \deg p = n \implies p$ has $n$ roots
where $p$ is some polynomial with coefficients in $\C$. | Every non-[[Definition:Constant Polynomial|constant]] [[Definition:Polynomial|polynomial]] of [[Definition:Degree of Polynomial|degree]] $n > 0$ with [[Definition:Polynomial Coefficient|coefficients]] in $\C$ has $n$ [[Definition:Root of Polynomial|roots]] in $\C$ including [[Definition:Multiplicity (Polynomial)|multip... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \mathbb N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\map \deg p = n \implies p$ has $n$ [[Definition:Root of Polynomial|roots]]
where $p$ is some [[Definition:Polynomial|polynomial]] with [[Definition:Polynomial Coe... | Number of Roots of Polynomial With Complex Coefficients Equals Degree of Polynomial | https://proofwiki.org/wiki/Number_of_Roots_of_Polynomial_With_Complex_Coefficients_Equals_Degree_of_Polynomial | https://proofwiki.org/wiki/Number_of_Roots_of_Polynomial_With_Complex_Coefficients_Equals_Degree_of_Polynomial | [
"Polynomial Theory",
"Algebra",
"Analysis",
"Fundamental Theorem of Algebra",
"Complex Numbers",
"Proofs by Induction"
] | [
"Definition:Constant Polynomial",
"Definition:Polynomial",
"Definition:Degree of Polynomial",
"Definition:Coefficient of Polynomial",
"Definition:Root of Polynomial",
"Definition:Multiplicity (Polynomial)"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Root of Polynomial",
"Definition:Polynomial",
"Definition:Coefficient of Polynomial",
"Definition:Proposition",
"Definition:Root of Polynomial",
"Definition:Polynomial",
"Definition:Root of Polynomial",
"Definition:Polyno... |
proofwiki-23494 | Linear Combination of Eigenvectors Corresponding to Same Eigenvalue of Linear Operator is Eigenvector | Let $K$ be a field.
Let $V$ be a vector space over $K$.
Let $\mathbf T : V \to V$ be a linear operator.
Let $\lambda \in K$ be an eigenvalue of $\mathbf T$ with geometric multiplicity $n$.
Let $\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n$ be eigenvectors corresponding to $\lambda$.
Let $\mathbf v$ be a non-zero li... | Applying $\mathbf T$ to $\mathbf v$ results in:
{{begin-eqn}}
{{eqn | l = \mathbf T \mathbf v
| r = \mathbf T \paren {\sum_{i \mathop = 1}^n c_i \mathbf v_i}
}}
{{eqn | r = \sum_{i \mathop = 1}^n c_i \mathbf T \mathbf v_i
| c = $\mathbf T$ is a linear operator
}}
{{eqn | r = \sum_{i \mathop = 1}^n c_i \lamb... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $V$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $\mathbf T : V \to V$ be a [[Definition:Linear Operator|linear operator]].
Let $\lambda \in K$ be an [[Definition:Eigenvalue of Linear Operator|eigenvalue]] of $\mathbf T$ with [[Definiti... | Applying $\mathbf T$ to $\mathbf v$ results in:
{{begin-eqn}}
{{eqn | l = \mathbf T \mathbf v
| r = \mathbf T \paren {\sum_{i \mathop = 1}^n c_i \mathbf v_i}
}}
{{eqn | r = \sum_{i \mathop = 1}^n c_i \mathbf T \mathbf v_i
| c = $\mathbf T$ is a [[Definition:Linear Operator|linear operator]]
}}
{{eqn | r = \... | Linear Combination of Eigenvectors Corresponding to Same Eigenvalue of Linear Operator is Eigenvector | https://proofwiki.org/wiki/Linear_Combination_of_Eigenvectors_Corresponding_to_Same_Eigenvalue_of_Linear_Operator_is_Eigenvector | https://proofwiki.org/wiki/Linear_Combination_of_Eigenvectors_Corresponding_to_Same_Eigenvalue_of_Linear_Operator_is_Eigenvector | [
"Eigenvalues of Linear Operators",
"Eigenvectors of Linear Operators",
"Linear Operators"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Linear Operator",
"Definition:Eigenvalue/Linear Operator",
"Definition:Geometric Multiplicity",
"Definition:Eigenvector/Linear Operator",
"Definition:Zero Vector",
"Definition:Linear Combination",
"Definition:Eigenvector/L... | [
"Definition:Linear Operator",
"Definition:Eigenvalue/Linear Operator",
"Indexed Summation of Multiple of Mapping",
"Definition:Eigenvector/Linear Operator",
"Definition:Eigenvalue/Linear Operator"
] |
proofwiki-23495 | Weakly Sigma-Locally Compact Space is Hemicompact | Let $T = \struct {S, \tau}$ be a weakly $\sigma$-locally compact space.
Then $T$ is a hemicompact space. | {{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
{{:Definition:Weakly Sigma-Locally Compact Space}}
{{Recall|Hemicompact Space|hemicompact space}}
{{:Definition:Hemicompact Space}}
{{ProofWanted}} | Let $T = \struct {S, \tau}$ be a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space]].
Then $T$ is a [[Definition:Hemicompact Space|hemicompact space]]. | {{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
{{:Definition:Weakly Sigma-Locally Compact Space}}
{{Recall|Hemicompact Space|hemicompact space}}
{{:Definition:Hemicompact Space}}
{{ProofWanted}} | Weakly Sigma-Locally Compact Space is Hemicompact | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_Space_is_Hemicompact | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_Space_is_Hemicompact | [
"Weakly Sigma-Locally Compact Spaces",
"Hemicompact Spaces"
] | [
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:Hemicompact Space"
] | [] |
proofwiki-23496 | Extremally Disconnected and Locally Hausdorff Space is Sequentially Discrete | Let $T = \struct {S, \tau}$ be an extremally disconnected space which is also locally Hausdorff.
Then $T$ is sequentially discrete. | {{Recall|Sequentially Discrete Space|sequentially discrete topological space}}
{{:Definition:Sequentially Discrete Space}}
Let $T = \struct {S, \tau}$ be an extremally disconnected space which is also locally Hausdorff.
{{Recall|Extremally Disconnected Space|extremally disconnected space}}
{{:Definition:Extremally Disc... | Let $T = \struct {S, \tau}$ be an [[Definition:Extremally Disconnected Space|extremally disconnected space]] which is also [[Definition:Locally Hausdorff Space|locally Hausdorff]].
Then $T$ is [[Definition:Sequentially Discrete Space|sequentially discrete]]. | {{Recall|Sequentially Discrete Space|sequentially discrete topological space}}
{{:Definition:Sequentially Discrete Space}}
Let $T = \struct {S, \tau}$ be an [[Definition:Extremally Disconnected Space|extremally disconnected space]] which is also [[Definition:Locally Hausdorff Space|locally Hausdorff]].
{{Recall|Extre... | Extremally Disconnected and Locally Hausdorff Space is Sequentially Discrete | https://proofwiki.org/wiki/Extremally_Disconnected_and_Locally_Hausdorff_Space_is_Sequentially_Discrete | https://proofwiki.org/wiki/Extremally_Disconnected_and_Locally_Hausdorff_Space_is_Sequentially_Discrete | [
"Extremally Disconnected Spaces",
"Locally Hausdorff Spaces",
"Sequentially Discrete Spaces"
] | [
"Definition:Extremally Disconnected Space",
"Definition:Locally Hausdorff Space",
"Definition:Sequentially Discrete Space"
] | [
"Definition:Extremally Disconnected Space",
"Definition:Locally Hausdorff Space"
] |
proofwiki-23497 | Hermitian Operator is Normal | All Hermitian operators are normal operators. | Let $\mathbf T$ be a Hermitian operator.
Then,
{{begin-eqn}}
{{eqn | l = \mathbf T^* \mathbf T
| r = \mathbf T \mathbf T
| c = {{Defof|Hermitian Operator}}
}}
{{eqn | r = \mathbf T \mathbf T^*
| c = {{Defof|Hermitian Operator}}
}}
{{end-eqn}}
Thus, $\mathbf T$ is a normal operator.
{{qed}}
Category:He... | All [[Definition:Hermitian Operator|Hermitian operators]] are [[Definition:Normal Operator|normal operators]]. | Let $\mathbf T$ be a [[Definition:Hermitian Operator|Hermitian operator]].
Then,
{{begin-eqn}}
{{eqn | l = \mathbf T^* \mathbf T
| r = \mathbf T \mathbf T
| c = {{Defof|Hermitian Operator}}
}}
{{eqn | r = \mathbf T \mathbf T^*
| c = {{Defof|Hermitian Operator}}
}}
{{end-eqn}}
Thus, $\mathbf T$ is a ... | Hermitian Operator is Normal | https://proofwiki.org/wiki/Hermitian_Operator_is_Normal | https://proofwiki.org/wiki/Hermitian_Operator_is_Normal | [
"Hermitian Operators",
"Normal Operators"
] | [
"Definition:Hermitian Operator",
"Definition:Normal Operator"
] | [
"Definition:Hermitian Operator",
"Definition:Normal Operator",
"Category:Hermitian Operators",
"Category:Normal Operators"
] |
proofwiki-23498 | Topological Space with Countable K-Network has Countable Network | Let $T = \struct {S, \tau}$ be a topological space which has a countable $k$-network.
Then $T$ has a countable network. | Follows directly from K-Network is Network.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which has a [[Definition:Countable Set|countable]] [[Definition:K-Network|$k$-network]].
Then $T$ has a [[Definition:Countable Set|countable]] [[Definition:Network (Topology)|network]]. | Follows directly from [[K-Network is Network]].
{{qed}} | Topological Space with Countable K-Network has Countable Network | https://proofwiki.org/wiki/Topological_Space_with_Countable_K-Network_has_Countable_Network | https://proofwiki.org/wiki/Topological_Space_with_Countable_K-Network_has_Countable_Network | [
"K-Networks",
"Networks (Topology)",
"Countable Sets"
] | [
"Definition:Topological Space",
"Definition:Countable Set",
"Definition:Network (Topology)/K-Network",
"Definition:Countable Set",
"Definition:Network (Topology)"
] | [
"K-Network is Network"
] |
proofwiki-23499 | K-Network is Network | Let $\NN$ be a $k$-network on a topological space $T = \struct {S, \tau}$.
Then $\NN$ is also a network on $T$. | {{Recall|K-Network|$k$-network}}
{{:Definition:K-Network/K-Network}}
{{Recall|Network (Topology)|network}}
{{:Definition:Network (Topology)/Definition 1}}
Let $T = \struct {S, \tau}$ have a $k$-network $\NN$.
Let $U \in \tau$ be arbitrary.
From Empty Set is Compact, $\O$ is a compact space.
From Empty Set is Subset of ... | Let $\NN$ be a [[Definition:K-Network|$k$-network]] on a [[Definition:Topological Space|topological space]] $T = \struct {S, \tau}$.
Then $\NN$ is also a [[Definition:Network (Topology)|network]] on $T$. | {{Recall|K-Network|$k$-network}}
{{:Definition:K-Network/K-Network}}
{{Recall|Network (Topology)|network}}
{{:Definition:Network (Topology)/Definition 1}}
Let $T = \struct {S, \tau}$ have a [[Definition:K-Network|$k$-network]] $\NN$.
Let $U \in \tau$ be [[Definition:Arbitrary|arbitrary]].
From [[Empty Set is Compac... | K-Network is Network | https://proofwiki.org/wiki/K-Network_is_Network | https://proofwiki.org/wiki/K-Network_is_Network | [
"K-Networks",
"Networks (Topology)"
] | [
"Definition:Network (Topology)/K-Network",
"Definition:Topological Space",
"Definition:Network (Topology)"
] | [
"Definition:Network (Topology)/K-Network",
"Definition:Arbitrary",
"Empty Set is Compact",
"Definition:Compact Topological Space",
"Empty Set is Subset of All Sets",
"Definition:Network (Topology)/K-Network",
"Definition:Subset",
"Definition:Arbitrary",
"Definition:Network (Topology)"
] |
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