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If $S$ is a nonempty open connected subset of a topological space $T$, and if every point of $T$ that is a limit point of $S$ is in $S$, then $S = T$. |
If $S$ is an open connected set, $w \in S$, $Y \subseteq X$, $X$ is a set of holomorphic functions on $S$, $X$ contains no functions that are identically $0$ or $1$, and $Y$ contains no functions that are bounded by $1$ at $w$, then there exists a constant $B$ such that for all $h \in Y$ and $z \in K$, we have $|h(z)| ... |
If $f$ is holomorphic on the punctured disk of radius $k$ and if $f$ is bounded on every punctured disk of radius $e$ with $0 < e < k$, then $f$ is bounded on the punctured disk of radius $\epsilon$ with $0 < \epsilon < k$. |
If $f$ is a holomorphic function on the punctured unit disk and $f(z) \neq 0$ and $f(z) \neq 1$ for all $z$ in the punctured unit disk, then there exists a positive number $e$ such that either $|f(z)| \leq B$ for all $z$ in the punctured disk of radius $e$ or $|f(z)| \geq B$ for all $z$ in the punctured disk of radius ... |
If $f$ is holomorphic on an open set $M$ and $f(w) \neq 0$ and $f(w) \neq a$ for all $w \in M - \{z\}$, then either $f$ or $1/f$ is bounded on a punctured neighborhood of $z$. |
If $f$ is holomorphic on an open set $M$ and $f$ does not take on the values $a$ and $b$, then either $f$ or $1/f$ has a limit at $z$. |
If $f$ is a holomorphic function on an open set $M$ that does not have a limit at $z \in M$, then there exists a point $a$ such that $f$ is surjective on $M - \{z\}$ except possibly at $a$. |
If $f$ is a holomorphic function on an open set $M$ that does not have a limit at $z \in M$, then there exists a point $a \in \mathbb{C}$ such that the set $\{x \in M - \{z\} : f(x) = a\}$ is infinite. |
Suppose $f$ is a holomorphic function defined on an open set $M$ that does not have a limit at $z \in M$. Then the image of $f$ is dense in $\mathbb{C}$. |
The interval $[0,1]$ is the union of the intervals $[0,1/2]$ and $[1/2,1]$. |
The interval $[0,1]$ is the union of the intervals $[0,1/2]$ and $[1/2,1]$. |
The image of a set $A$ under the function $x \mapsto (x, c)$ is equal to the Cartesian product of $A$ and the singleton set $\{c\}$. |
The composition of the function $(x,y) \mapsto (f(x,y), g(x,y))$ with the function $(x,y) \mapsto x$ is the function $(x,y) \mapsto f(x,y)$. |
$\mathrm{snd} \circ (x,y) \mapsto (f(x,y), g(x,y)) = (x,y) \mapsto g(x,y)$. |
If $h$ is a continuous function from $T \times X$ to $Y$, then for any $t \in T$, the function $h(t, \cdot)$ is continuous from $X$ to $Y$. |
If $h$ is a continuous map from $X \times Y$ to $Z$, then for any $t \in X$, the map $h(t, \cdot)$ is a continuous map from $Y$ to $Z$. |
Two continuous maps $p, q: X \to Y$ are homotopic if and only if there exists a continuous map $h: [0,1] \times X \to Y$ such that $h(0,x) = p(x)$ and $h(1,x) = q(x)$ for all $x \in X$. |
If $f$ and $g$ are homotopic maps from $X$ to $Y$ and $P$ is a property of maps from $X$ to $Y$ such that any continuous map from $X$ to $Y$ with property $P$ also has property $Q$, then $f$ and $g$ are homotopic with property $Q$. |
If two maps are homotopic, then they are both continuous. |
If two maps $f$ and $g$ are homotopic, then they are both continuous. |
If two maps $f$ and $g$ are homotopic with a property $P$, then $f$ and $g$ have property $P$. |
If two continuous maps $f$ and $g$ are equal on the topological space $X$, then they are homotopic. |
If two maps $f$ and $g$ are homotopic, then the image of $f$ is a subset of the codomain of $g$. |
If $f$ and $g$ are homotopic maps from $X$ to $Y$, then $g(X) \subseteq Y$. |
If $f$ and $g$ are homotopic maps from $X$ to $Y$ with property $P$, then they are also homotopic maps from $Z$ to $Y$ with property $P$, provided $Z$ is a subset of $X$. |
If $f$ and $g$ are homotopic with respect to a property $P$ on $X$ and $Y$, and $Y$ is a subset of $Z$, then $f$ and $g$ are homotopic with respect to $P$ on $X$ and $Z$. |
A continuous map $f$ is homotopic to itself if and only if $f$ is continuous. |
If $f$ and $g$ are homotopic, then $g$ and $f$ are homotopic. |
Two continuous maps $f, g: X \to Y$ are homotopic if and only if $g, f: X \to Y$ are homotopic. |
If $f$ and $g$ are homotopic and $g$ and $h$ are homotopic, then $f$ and $h$ are homotopic. |
If $g \circ f$ is the identity on the topological space $X$, then $g \circ f$ is homotopic to the identity map on $X$. |
If $f$ and $g$ are homotopic maps from $X_1$ to $X_2$ and $h$ is a continuous map from $X_2$ to $X_3$, then $h \circ f$ and $h \circ g$ are homotopic maps from $X_1$ to $X_3$. |
If $f$ and $g$ are homotopic maps from $X_2$ to $X_3$, and $h$ is a continuous map from $X_1$ to $X_2$, then $f \circ h$ and $g \circ h$ are homotopic maps from $X_1$ to $X_3$. |
If $f$ and $f'$ are homotopic maps from $X$ to $Y$ and $g$ and $g'$ are homotopic maps from $Y$ to $Z$, then $g \circ f$ and $g' \circ f'$ are homotopic maps from $X$ to $Z$. |
If $f$ and $g$ are homotopic maps from $X$ to $Y$ and $h$ is a continuous map from $W$ to $X$ such that $h(W) \subseteq X$, then $f \circ h$ and $g \circ h$ are homotopic maps from $W$ to $Y$. |
If $f$ and $g$ are homotopic maps from $X$ to $Y$ and $h$ is a continuous map from $Y$ to $Z$ such that $h(Y) \subseteq Z$, then $h \circ f$ and $h \circ g$ are homotopic maps from $X$ to $Z$. |
If two maps are homotopic on a subspace, then they are homotopic on the whole space. |
If $X$ is empty, then any two maps $f, g: X \to X'$ are homotopic. |
If $X$ is empty, then $f$ and $g$ are homotopic if and only if $f$ and $g$ are both continuous. |
The identity map on the empty set is homotopic to any other map. |
Two constant maps are homotopic if and only if they map to the same path component. |
If two continuous maps $f$ and $g$ are homotopic, then so are $f'$ and $g'$ if $f'$ and $g'$ are equal to $f$ and $g$ on the topological space $X$. |
If $f$ and $g$ are homotopic maps from $X_1$ to $Y_1$ and $X_2$ to $Y_2$, respectively, then the map $(x,y) \mapsto (f(x),g(y))$ is homotopic to the map $(x,y) \mapsto (f'(x),g'(y))$. |
If $f_i$ and $g_i$ are homotopic maps from $X_i$ to $Y_i$ for each $i \in I$, then the product map $(f_i)_{i \in I}$ is homotopic to the product map $(g_i)_{i \in I}$. |
Two continuous maps $f, g: S \to T$ are homotopic if and only if $S$ is empty or $T$ is path-connected and $f$ is homotopic to a constant map. |
Two paths $p$ and $q$ are homotopic if and only if there exists a continuous function $h$ from the unit square to $S$ such that $h$ maps the boundary of the unit square to $p$ and $q$ and $h$ maps the interior of the unit square to $S$. |
If two paths are homotopic, then they have the same starting point. |
If two paths are homotopic, then they have the same endpoints. |
If two paths are homotopic, then they are both paths. |
If two paths are homotopic, then their images are contained in the same set. |
Two paths are homotopic if and only if they are the same path. |
If two paths are homotopic, then they are homotopic in the opposite direction. |
Two paths $p$ and $q$ are homotopic if and only if $q$ and $p$ are homotopic. |
If two paths are homotopic and a third path is homotopic to the second, then the third path is homotopic to the first. |
If $p$ is a path in $s$ and $q$ is a path in $s$ such that $p(t) = q(t)$ for all $t \in [0,1]$, then $p$ and $q$ are homotopic paths in $s$. |
If $p$ is a path in $s$ and $f$ is a continuous function from $[0,1]$ to $[0,1]$ such that $f(0) = 0$ and $f(1) = 1$, then the path $q$ defined by $q(t) = p(f(t))$ is homotopic to $p$. |
If two paths are homotopic in a subset $s$ of a topological space $t$, then they are homotopic in $t$. |
Suppose $p$ and $q$ are continuous functions from $[0,1] \times [0,1]$ to a topological space $X$. If $p(t)$ and $q(t)$ are paths in $X$ such that $p(t)$ ends at the same point as $q(t)$ starts for each $t \in [0,1]$, then the function $y \mapsto (p(x) +++ q(x))(y)$ is continuous. |
If two paths are homotopic, then their reverses are homotopic. |
The paths $p$ and $q$ are homotopic if and only if the paths $p^{-1}$ and $q^{-1}$ are homotopic. |
If two paths $p$ and $q$ are homotopic, and two paths $p'$ and $q'$ are homotopic, and the endpoints of $p$ and $q$ are the same, then the concatenation of $p$ and $q$ is homotopic to the concatenation of $p'$ and $q'$. |
If $f$ and $g$ are homotopic paths in $s$, and $h$ is a continuous map from $s$ to $t$, then $h \circ f$ and $h \circ g$ are homotopic paths in $t$. |
If $p$ is a path in $s$, then $p$ is homotopic to $p$ followed by a constant path. |
If $p$ is a path in $s$, then $p$ is homotopic to the concatenation of the constant path at the start of $p$ with $p$. |
If $p$, $q$, and $r$ are paths in a topological space $X$ such that the endpoints of $p$ and $q$ coincide, and the endpoints of $q$ and $r$ coincide, then the paths $p + (q + r)$ and $(p + q) + r$ are homotopic. |
If $p$ is a path in $s$, then $p$ followed by the reverse of $p$ is homotopic to a constant path. |
If $p$ is a path in $s$, then the path $p$ followed by the reverse of $p$ is homotopic to the constant path at the endpoint of $p$. |
Two loops $p$ and $q$ are homotopic if and only if there exists a continuous function $h$ from the unit square to $S$ such that $h(0,x) = p(x)$, $h(1,x) = q(x)$, and $h(t,0) = h(t,1)$ for all $t \in [0,1]$. |
If two paths are homotopic loops, then they are loops. |
If two loops are homotopic, then they are both paths. |
If two loops are homotopic, then their images are contained in the same set. |
A loop $p$ is homotopic to itself if and only if $p$ is a path with image in $s$ and $p$ is closed. |
If two loops are homotopic, then they are homotopic in the opposite direction. |
Two loops $p$ and $q$ are homotopic if and only if $q$ and $p$ are homotopic. |
If two loops are homotopic, and a third loop is homotopic to the second, then the third loop is homotopic to the first. |
If two loops are homotopic in a space $s$, then they are homotopic in any space $t$ containing $s$. |
If two paths $p$ and $q$ have the same image and the same endpoints, then they are homotopic loops. |
If two loops are homotopic, then their continuous images are homotopic. |
If two paths are homotopic and have the same starting and ending points, then they are homotopic loops. |
If two loops are homotopic, then the paths that they trace out are homotopic. |
If $p$ and $q$ are paths in a set $s$ such that $p$ ends where $q$ starts, then the loop $pqp^{-1}$ is homotopic to $q$. |
If $p$ and $q$ are paths in $S$ such that $p$ followed by the reverse of $q$ is homotopic to a constant path, then $p$ is homotopic to $q$. |
If $f$ and $g$ are continuous functions from $S$ to $\mathbb{R}^n$ and the line segment between $f(x)$ and $g(x)$ is contained in $t$ for all $x \in S$, then $f$ and $g$ are homotopic relative to $S$ with $t$ as the target space. |
If $g$ and $h$ are paths with the same endpoints, and the line segments between $g(t)$ and $h(t)$ are contained in $S$ for all $t$, then $g$ and $h$ are homotopic in $S$. |
If $g$ and $h$ are paths with the same endpoints, and the line segments from $g(t)$ to $h(t)$ are contained in $S$ for all $t$, then $g$ and $h$ are homotopic loops in $S$. |
If $g$ and $h$ are paths with the same endpoints, and $h$ is closer to $g$ than to any point outside $S$, then $g$ and $h$ are homotopic in $S$. |
If $g$ and $h$ are paths with the same endpoints, and $h$ is closer to $g$ than to any point outside $S$, then $g$ and $h$ are homotopic loops. |
If $g$ is a path in an open set $S$, then there is an $\epsilon > 0$ such that any path $h$ that is close enough to $g$ is homotopic to $g$ in $S$. |
If $g$ is a closed path in an open set $S$, then there is an $\epsilon > 0$ such that any closed path $h$ that is $\epsilon$-close to $g$ is homotopic to $g$ in $S$. |
If $g$ is a path in $s$ and $u, v, w$ are points in $[0, 1]$ such that $u \leq v \leq w$, then the concatenation of the subpaths of $g$ from $u$ to $v$ and from $v$ to $w$ is homotopic to the subpath of $g$ from $u$ to $w$. |
If $g$ is a path from $u$ to $v$ and $h$ is a path from $v$ to $w$, then the path from $u$ to $w$ obtained by concatenating $g$ and $h$ is homotopic to the path from $w$ to $u$ obtained by concatenating $h$ and $g$. |
If $g$ is a path in $s$ and $u, v, w$ are points in $[0, 1]$ such that $g$ is homotopic to the concatenation of the subpaths $g_{[u, v]}$ and $g_{[v, w]}$, then $g$ is homotopic to the concatenation of the subpaths $g_{[v, w]}$ and $g_{[w, u]}$. |
If $g$ is a path in $s$ and $u, v, w \in [0, 1]$, then the concatenation of the subpaths $g_{[u, v]}$ and $g_{[v, w]}$ is homotopic to the subpath $g_{[u, w]}$. |
If $a$ and $b$ are in the same path component of $S$, then the constant loops at $a$ and $b$ are homotopic. |
If two loops are homotopic, then they have the same value at any point in the interval $[0,1]$. |
Two points $a$ and $b$ are in the same path component of $S$ if and only if there is a homotopy between the constant loops at $a$ and $b$. |
A topological space $S$ is path-connected if and only if for every pair of points $a, b \in S$, the constant loops at $a$ and $b$ are homotopic. |
The empty set is simply connected. |
If a set $S$ is simply connected, then it is path connected. |
If $S$ is simply connected, then $S$ is connected. |
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