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A set $S$ is simply connected if and only if for every path $p$ with image in $S$ and every point $a \in S$, the path $p$ is homotopic to the constant path at $a$ in $S$. |
A set $S$ is simply connected if and only if it is path connected and every loop in $S$ is homotopic to a constant loop. |
A set $S$ is simply connected if and only if $S$ is empty or there exists a point $a \in S$ such that for every loop $p$ in $S$ with $p(0) = p(1) = a$, $p$ is homotopic to the constant loop at $a$. |
A set $S$ is simply connected if and only if it is path connected and every path in $S$ with the same start and end point is homotopic to a constant path. |
A set $S$ is simply connected if and only if it is path connected and any two paths with the same endpoints are homotopic. |
The product of two simply connected sets is simply connected. |
If $S$ is contractible, then $S$ is simply connected. |
If a set is contractible, then it is connected. |
If $S$ is contractible, then $S$ is path-connected. |
If $f: S \to T$ and $g: T \to U$ are continuous maps, and $T$ is contractible, then $g \circ f$ is nullhomotopic. |
If $f$ is a continuous map from $S$ to $T$ and $T$ is contractible, then $f$ is nullhomotopic. |
If $S$ is contractible and $f: S \to T$ is continuous, then $f$ is nullhomotopic. |
If $f_1$ and $f_2$ are continuous maps from $S$ to $T$, and $g_1$ and $g_2$ are continuous maps from $T$ to $U$, and $T$ is contractible and $U$ is path-connected, then $g_1 \circ f_1$ and $g_2 \circ f_2$ are homotopic. |
If $f$ and $g$ are continuous maps from $S$ to $T$, where $T$ is contractible, then $f$ and $g$ are homotopic. |
If $S$ is contractible and $T$ is path-connected, then any two continuous maps from $S$ to $T$ are homotopic. |
The whole complex plane is starlike. |
If $S$ is a nonempty convex set, then $S$ is starlike. |
If $S$ is a nonempty convex set and $T$ is a set such that the relative interior of $S$ is a subset of $T$ and $T$ is a subset of the closure of $S$, then $T$ is starlike. |
If $S$ is a starlike set, then $S$ is contractible. |
If $S$ is a starlike set, then $S$ is contractible. |
The whole space $\mathbb{R}^n$ is contractible. |
If $S$ is starlike, then $S$ is simply connected. |
Any convex set is simply connected. |
If $S$ is starlike, then $S$ is path-connected. |
Any starlike set is connected. |
A set $S$ is an interval if and only if it is simply connected. |
The empty space is contractible. |
If $S$ is a convex set and $T$ is a subset of $S$ that contains the relative interior of $S$ and is contained in the closure of $S$, then $T$ is contractible. |
Any convex set is contractible. |
The singleton set $\{a\}$ is contractible. |
A real interval is contractible. |
If $S$ and $T$ are contractible sets, then $S \times T$ is contractible. |
If for every open set $w$ and every point $x$ in $w$, there exists an open set $u$ containing $x$ and a set $v$ such that $P(v)$ holds and $u \subseteq v \subseteq w$, then $P$ holds locally on $S$. |
If $S$ is locally $P$, then for any open set $w$ in $S$ and any point $x$ in $w$, there exists an open set $u$ in $S$ containing $x$ and a set $v$ containing $u$ such that $v$ is contained in $w$ and $v$ is $P$. |
If $S$ is locally $P$, then $S$ is locally $Q$ whenever $Q$ implies $P$. |
If $S$ is locally $P$ and $t$ is an open subset of $S$, then $t$ is locally $P$. |
If $S$ is locally $P$ and $t$ is closed in $S$, then $S - t$ is locally $P$. |
The empty set is locally empty. |
A set is locally $P$ if and only if it is $P$. |
A set $S$ is locally $P$ if and only if for every open set $T$ containing a point $x$ of $S$, there exists an open set $U$ containing $x$ such that $S \cap U$ is contained in a set $V$ satisfying $P$ and contained in $S \cap T$. |
If $S$ and $T$ are locally $P$, then $S \cap T$ is locally $P$. |
If $S$ and $T$ are locally $P$ and $Q$, respectively, then $S \times T$ is locally $R$. |
If $f$ is a homeomorphism from a locally $P$ space $S$ to a space $T$, and if $Q$ is a property such that whenever $S$ is a $P$ space and $f$ is a homeomorphism from $S$ to $S'$, then $S'$ has property $Q$, then $T$ has property $Q$. |
If $f$ is a homeomorphism between $S$ and $T$, and $P$ and $Q$ are properties of topological spaces such that $S$ has property $P$ if and only if $T$ has property $Q$, then $S$ is locally $P$ if and only if $T$ is locally $Q$. |
If $S$ and $T$ are homeomorphic, then $S$ is locally $P$ if and only if $T$ is locally $Q$. |
If two topological spaces are homeomorphic, then they are locally compact if and only if the other is locally compact. |
If $P$ is a property of sets that is invariant under translations, then the property of being locally $P$ is also invariant under translations. |
If $f$ is a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ that is injective, then $f$ is a local homeomorphism. |
If $f$ is a continuous map from a topological space $S$ to a topological space $T$, and if $f$ maps open sets in $S$ to open sets in $T$, then $f$ is an open map. |
If $S$ is a connected set, $P$ and $Q$ are properties of points of $S$, and $a$ and $b$ are points of $S$ such that $P$ holds for $a$ and $b$ and $Q$ holds for $a$, then $Q$ holds for $b$. |
If $S$ is a connected set, $a, b \in S$, $P(a)$ and $P(b)$ are true, and $R$ is a relation on $S$ that is transitive, symmetric, and reflexive, then $R(a, b)$ is true. |
Suppose $S$ is a connected set and $P$ is a property that holds for $a \in S$. If $P$ is preserved by open neighborhoods, then $P$ holds for all $b \in S$. |
If $S$ is a connected set and $R$ is a symmetric and transitive relation on $S$ such that for every $a \in S$, there exists an open set $T$ containing $a$ such that $R$ holds for every pair of points in $T$, then $R$ holds for every pair of points in $S$. |
If $f$ is locally constant on a connected set $S$, then $f$ is constant on $S$. |
If $S$ is connected, then $f$ is locally constant on $S$ if and only if $f$ is constant on $S$. |
A set $S$ is locally compact if and only if for every $x \in S$, there exist open sets $U$ and $V$ such that $x \in U \subseteq V \subseteq S$ and $U$ and $V$ are compact. |
If $S$ is a locally compact set, then for every $x \in S$, there exist an open set $u$ containing $x$ and a compact set $v$ containing $u$ such that $v \subseteq S$. |
A topological space $S$ is locally compact if and only if for every $x \in S$, there exists an open set $U$ containing $x$ such that $U$ is compact and contained in $S$. |
A topological space $X$ is locally compact if and only if for every $x \in X$, there exists an open ball $B(x, \epsilon)$ such that $B(x, \epsilon) \cap X$ is closed. |
A set $S$ is locally compact if and only if for every compact subset $K$ of $S$, there exist open subsets $U$ and $V$ of $S$ such that $K \subseteq U \subseteq V \subseteq S$ and $V$ is compact. |
Any open set is locally compact. |
If $S$ is a closed set, then $S$ is locally compact. |
The set of all real numbers is locally compact. |
If $S$ and $t$ are locally compact, then $S \cap t$ is locally compact. |
If $t$ is a closed subset of a locally compact set $S$, then $t$ is locally compact. |
If $S$ is a locally compact space, then $S - \{a\}$ is also locally compact. |
A set $S$ is locally closed if and only if it is locally compact. |
If $S$ and $T$ are locally compact, and $S$ and $T$ are open in $S \cup T$, then $S \cup T$ is locally compact. |
If $S$ and $T$ are locally compact and closed in $S \cup T$, then $S \cup T$ is locally compact. |
If $S$ and $T$ are locally compact, then $S \times T$ is locally compact. |
A topological space is locally compact if and only if every compact subset of it is contained in an open subset of it that is also compact. |
If $S$ is a compact set and $C$ is a component of $S$, then $C$ is the intersection of all open and closed subsets of $S$ that contain $C$. |
If $C$ is a compact component of a locally compact set $S$, then there exists a compact open set $K$ such that $C \subseteq K \subseteq U$. |
Suppose $S$ is a locally compact set, $C$ is a component of $S$, and $C$ is compact. Then there exists a compact open set $K$ such that $C \subseteq K \subseteq U$. |
If $S$ is a locally compact set, $C$ is a component of $S$, and $C$ is compact, then $C$ is the intersection of all compact open sets containing $C$. |
If every point in a topological space has a connected neighborhood, then the space is locally connected. |
If $S$ is locally connected and $t$ is an open subset of $S$, then the connected component of $t$ containing $x$ is open in $S$. |
If $S$ is a topological space and $v$ is an open subset of $S$, then for every $x \in v$, there exists an open connected subset $u$ of $S$ such that $x \in u \subseteq v$. |
A topological space $S$ is locally connected if and only if for every open set $v$ containing a point $x$, there exists an open set $u$ containing $x$ such that $u$ is connected and $u \subseteq v$. |
A topological space is locally connected if and only if every point in every open set is contained in an open connected set. |
If every point in $S$ has a neighborhood that is path-connected, then $S$ is locally path-connected. |
If $S$ is locally path-connected and $t$ is an open subset of $S$, then the path-component of $x$ in $t$ is open in $S$. |
If $S$ is locally path-connected, then for every open set $v$ and every point $x \in v$, there exists an open set $u$ such that $x \in u \subseteq v$ and $u$ is path-connected. |
A topological space $S$ is locally path-connected if and only if for every open set $V$ containing a point $x$, there exists an open set $U$ containing $x$ such that $U$ is path-connected and $U \subseteq V$. |
A topological space is locally path-connected if and only if every point has an open path-connected neighborhood. |
A topological space is locally connected if and only if every component of every open subset is open. |
A topological space is locally connected if and only if for every open set $U$ and every point $x \in U$, there exists an open set $V$ such that $x \in V \subseteq U$ and for every $y \in V$, there exists a connected set $C$ such that $x, y \in C \subseteq U$. |
A topological space is locally path-connected if and only if for every point $x$ and every open neighborhood $V$ of $x$, there exists an open neighborhood $U$ of $x$ such that for every point $y$ in $U$, there exists a path from $x$ to $y$ whose image is contained in $V$. |
If every point in a space $S$ has a neighborhood that is path-connected, then every point in $S$ has a neighborhood that is connected. |
If $S$ is a locally connected space and $c$ is a component of $S$, then $c$ is locally connected. |
If $S$ is locally path-connected and $c$ is a component of $S$, then $c$ is locally path-connected. |
If $S$ is locally path-connected, then the connected component of $S$ containing $x$ is also locally path-connected. |
Any open set is locally path-connected. |
Any open set is locally connected. |
The real vector space $\mathbb{R}^n$ is locally path-connected. |
The real line is locally connected. |
If $S$ is locally connected, then the connected component of $S$ containing $x$ is open in $S$. |
If $S$ is locally connected and $c$ is a component of $S$, then $c$ is open in $S$. |
If $S$ is locally path-connected, then the path-component of $x$ in $S$ is open in $S$. |
If $S$ is locally path-connected, then the path-component of $x$ in $S$ is closed in $S$. |
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