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Any convex set is locally path-connected. |
Any convex set is locally connected. |
If $S$ is locally path-connected, then the path-component and connected-component of a point $x$ in $S$ are the same. |
If $S$ is locally path-connected, then the path-component of $x$ is the same as the connected component of $x$. |
If $S$ is locally path-connected, then the path-component of $S$ containing $x$ is locally path-connected. |
If $S$ is an open set, then the path component of $x$ in $S$ is the same as the connected component of $x$ in $S$. |
If $S$ is an open set, then the path component of $x$ in $S$ is the same as the connected component of $x$ in $S$. |
If $f$ is a continuous surjective map from a locally connected space $S$ to a space $T$, then $T$ is locally connected. |
If $S$ is locally path-connected and $f$ is a quotient map from $S$ to $f(S)$, then $f(S)$ is locally path-connected. |
If $f$ is continuous on each component of $S$, then $f$ is continuous on $S$. |
If $f$ is continuous on each component of a locally connected space $S$, then $f$ is continuous on $S$. |
If $S$ is locally connected, then a function $f$ is continuous on $S$ if and only if it is continuous on each component of $S$. |
If $f$ is continuous on each component of an open set $S$, then $f$ is continuous on $S$. |
If $S$ is open, then $f$ is continuous on $S$ if and only if $f$ is continuous on each component of $S$. |
If $U$ is locally connected, $S$ is closed in $U$, and $c$ is a collection of components of $U - S$, then $S \cup \bigcup c$ is closed in $U$. |
If $S$ is a closed set and $c$ is a collection of components of the complement of $S$, then $S \cup \bigcup c$ is closed. |
If $u$ is locally connected, $S$ is closed in $u$, and $c$ is a component of $u - S$, then $S \cup c$ is closed in $u$. |
If $S$ is a closed set and $c$ is a component of the complement of $S$, then $S \cup c$ is closed. |
If $S$ is a subspace of $\mathbb{R}^n$ and $T$ is a subspace of $\mathbb{R}^m$ with $n \leq m$, then there exists a linear map $f: \mathbb{R}^n \to \mathbb{R}^m$ such that $f(S) \subseteq T$ and $||f(x)|| = ||x||$ for all $x \in S$. |
If two subspaces of Euclidean space have the same dimension, then there exist linear isometries between them. |
If $S$ and $T$ are subspaces of Euclidean spaces of the same dimension, then there exists a linear isometry from $S$ to $T$. |
If two Euclidean spaces have the same dimension, then there exists a linear isomorphism between them. |
Any two subspaces of Euclidean spaces of the same dimension are homeomorphic. |
If two affine sets have the same affine dimension, then they are homeomorphic. |
Suppose $f$ and $g$ are continuous maps from $U$ to $t$ such that $f$ and $g$ are homotopic in $s$ via a homotopy $H$ with the property that $H(x,0) = f(x)$ and $H(x,1) = g(x)$ for all $x \in U$. Then $f$ and $g$ are homotopic in $t$ via the homotopy $H'$ defined by $H'(x,t) = H(x,2t)$ for $t \in [0,1/2]$ and $H'(x,t) ... |
Suppose $f$ is a continuous function from a topological space $U$ to a topological space $t$. Suppose that $f$ is homotopic to a constant function $c$ in $U$ with respect to a property $P$. Suppose that $P$ is preserved by composition with continuous functions from $U$ to $s$. Suppose that $Q$ is a property that is pre... |
If $f$ and $g$ are continuous maps from $t$ to $U$ such that $f$ and $g$ are homotopic in $U$ and $f$ and $g$ are homotopic in $U$, then $f$ and $g$ are homotopic in $U$. |
If $f$ is a continuous map from a topological space $X$ to a topological space $Y$, and if $Y$ is compact, then $f$ is uniformly continuous. |
If $S$ is simply connected, $h$ is a continuous map from $S$ to $T$, $k$ is a continuous map from $T$ to $S$, and $k$ is a retraction of $h$, then $T$ is simply connected. |
If two spaces are homeomorphic and one of them is simply connected, then the other is also simply connected. |
If two spaces are homeomorphic, then they are simply connected if and only if the other is. |
If two topological spaces are homeomorphic, then they are homotopy equivalent. |
A space is homotopy equivalent to itself. |
Two spaces are homotopy equivalent if and only if their homotopy types are isomorphic. |
If $X$ is homotopy equivalent to $Y$ and $Y$ is homotopy equivalent to $U$, then $X$ is homotopy equivalent to $U$. |
If $X$ and $Y$ are topological spaces and $r$ and $s$ are continuous maps such that $r$ is a retraction and $s \circ r$ is homotopic to the identity map on $X$, then $X$ and $Y$ are homotopy equivalent. |
If $X$ is homotopy equivalent to $Y$, then $X$ is homotopy equivalent to $Y$. |
A subset $S$ of a topological space $X$ is a deformation retract of $X$ if and only if $S$ is a retract of $X$ and there exists a continuous map $f: X \to S$ such that $f$ is homotopic to the identity map on $X$. |
A set $S$ is contractible if and only if the topological space $(S, \tau)$ is contractible, where $\tau$ is the topology generated by the set $S$. |
The empty space is contractible. |
If $X$ is a topological space with a single point, then $X$ is contractible. |
If the topological space $X$ is a subset of a singleton, then $X$ is contractible. |
The subspace of a topological space consisting of a single point is contractible. |
A topological space $X$ is contractible if and only if it is empty or there exists a point $a \in X$ such that the identity map on $X$ is homotopic to the constant map at $a$. |
Any contractible space is path-connected. |
If $X$ is contractible, then $X$ is connected. |
A topological space $X$ is contractible if and only if for every point $a \in X$, the identity map $X \to X$ is homotopic to the constant map $X \to X$ that maps every point to $a$. |
The composition of a constant function with another function is a constant function. |
If $f: X \to Y$ and $g: Y \to Z$ are continuous maps and $Y$ is a contractible space, then $g \circ f$ is nullhomotopic. |
If $f$ is a continuous map from $X$ to a contractible space $Y$, then $f$ is nullhomotopic. |
If $X$ is a contractible space and $f: X \to Y$ is a continuous map, then $f$ is nullhomotopic. |
If $X$ is contractible and $f: X \to Y$ and $g: Y \to X$ are continuous maps such that $f \circ g$ is homotopic to the identity map on $Y$, then $Y$ is contractible. |
If two topological spaces are homotopy equivalent, then they are both contractible or both not contractible. |
If two topological spaces are homeomorphic, then they are contractible if and only if the other is contractible. |
A topological space $X$ is contractible if and only if it is empty or it is homotopy equivalent to a singleton. |
If $X$ is a contractible space and $f: X \to Y$ is a retraction map, then $Y$ is a contractible space. |
A product of two topological spaces is contractible if and only if one of the spaces is empty or both spaces are contractible. |
A product of contractible spaces is contractible. |
A subspace of the real line is contractible if and only if it is an interval. |
The Euclidean real numbers are contractible. |
If two topological spaces are homeomorphic, then they are homotopy equivalent. |
If $f$ is a linear injection, then $f(S)$ is homotopy equivalent to $S$. |
The translation of a set by a vector is homotopy equivalent to the original set. |
If two spaces $S$ and $T$ are homotopy equivalent, and $f$ and $g$ are continuous maps from $U$ to $T$ such that $f$ and $g$ are homotopic in $S$, then $f$ and $g$ are homotopic in $T$. |
If two topological spaces are homotopy equivalent, then a map from a third space to the first space is homotopic to a map to the second space if and only if the two maps are homotopic. |
If two spaces are homotopy equivalent, then any continuous map from one to a third space is homotopic to a constant map. |
If $S$ is homotopy equivalent to $T$, then a continuous map $f: S \to U$ is null-homotopic if and only if the same is true for the map $f: T \to U$. |
If $S$ and $T$ are homotopy equivalent, and $f$ is a continuous map from $U$ to $T$ such that any continuous map from $U$ to $S$ is homotopic to a constant map, then $f$ is homotopic to a constant map. |
If two topological spaces are homotopy equivalent, then a continuous map from a third space to the first space is null-homotopic if and only if the same map is null-homotopic when the first space is replaced by the second space. |
If two sets are contractible, then they are homotopy equivalent. |
A set $S$ is homotopy equivalent to the empty set if and only if $S$ is empty. |
Two sets are homotopy equivalent if and only if they are both empty. |
If two topological spaces are homotopy equivalent, then they are contractible if and only if they are contractible. |
A set $S$ is homotopy equivalent to a single point if and only if $S$ is nonempty and contractible. |
If two topological spaces are homeomorphic, then they are contractible if and only if they are contractible. |
If $S$ is contractible and homeomorphic to $T$, then $T$ is contractible. |
If $S$ is a bounded set and $-S$ is connected, then $S$ is empty. |
If $S$ is a bounded set and the complement of $S$ is path-connected, then $S$ is empty. |
If a set $S$ is bounded and its complement is connected, then $S$ is empty. |
If $a \neq b$, then the closed segment $[a,b]$ is uncountable. |
If $a \neq b$, then the open segment between $a$ and $b$ is uncountable. |
If $S$ is a convex set containing two distinct points $a$ and $b$, then $S$ is uncountable. |
The ball of radius $r$ around $a$ is uncountable. |
If $A$ is a countable set in a Euclidean space, then the ball of radius $r$ around $z$ minus $A$ is nonempty for any $r > 0$. |
The open ball of radius $r$ around $a$ is uncountable. |
If the sets in a collection $\mathcal{N}$ are pairwise disjoint, then $\mathcal{N}$ is countable. |
If the union of a collection of pairwise disjoint sets is countable, then the collection is countable. |
If $S$ is a connected set containing two distinct points $a$ and $b$, then $S$ is uncountable. |
If $S$ is a path-connected set containing two distinct points $a$ and $b$, then $S$ is uncountable. |
A connected set is finite if and only if it is empty or a singleton. |
If $S$ is a connected uncountable set, then $S$ is not a singleton. |
If $g$ is a simple path, then the image of $g$ is uncountable. |
If $g$ is an arc, then the image of $g$ is uncountable. |
If $U$ is a convex set in $\mathbb{R}^n$ that is not contained in a line, and $S$ is a countable set, then $U - S$ is path-connected. |
If $U$ is a convex set in $\mathbb{R}^n$ that is not contained in a hyperplane, and $S$ is a countable set, then $U - S$ is connected. |
If $S$ is a convex set and $\operatorname{affdim}(S) \neq 1$, then $S - \{a\}$ is path-connected for any $a \in S$. |
If $S$ is a convex set of dimension at least 2, then $S - \{a\}$ is connected for any $a \in S$. |
If $S$ is a countable set in $\mathbb{R}^n$ with $n \geq 2$, then $\mathbb{R}^n \setminus S$ is path-connected. |
If $S$ is a connected set that is open in its affine hull and $T$ is a countable set, then $S - T$ is path-connected. |
If $S$ is a connected set that is open in its affine hull and $T$ is a countable set, then $S - T$ is connected. |
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