Category:Path-Connected Sets

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This category contains results about Path-Connected Sets in the context of Topology.

Let $T = \struct {S, \tau}$ be a topological space.

Let $U \subseteq S$ be a subset of $S$.

Let $T' = \struct {U, \tau_U}$ be the subspace of $T$ induced by $U$.


Then $U$ is a path-connected set in $T$ if and only if every two points in $U$ are path-connected in $T\,'$.


That is, $U$ is a path-connected set in $T$ if and only if:

for every $x, y \in U$, there exists a continuous mapping $f: \closedint 0 1 \to U$ such that:
$\map f 0 = x$
and:
$\map f 1 = y$