Category:Path-Connected Sets

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


Let $T = \left({S, \tau}\right)$ be a topological space.

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

Let $T' = \left({U, \tau_U}\right)$ 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 $f: \left[{0 \,.\,.\, 1}\right] \to U$ such that $f \left({0}\right) = x$ and $f \left({1}\right) = y$.