Locally Path-Connected iff Path Components of Open Subsets are Open

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Theorem

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


Then $T = \left({S, \tau}\right)$ is locally path-connected if and only if the path components of open sets of $T$ are also open in $T$.


Proof

Necessary Condition

Follows directly from:

Open Subset of Locally Path-Connected Space is Locally Path-Connected
Path Component of Locally Path-Connected Space is Open
Open Set in Open Subspace

$\blacksquare$


Sufficient Condition


Also see