Path Components are Open iff Union of Open Path-Connected Sets/Path Components are Open implies Space is Union of Open Path-Connected Sets

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Theorem

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

Let the path components of $T$ be open sets.

Then:

$S$ is a union of open path-connected sets of $T$.

Proof

By definition, the path components of $T$ are a partition of $S$.

Hence $S$ is the union of the open path components of $T$.

Since a path component is a maximal path-connected set by definition, then $S$ is a union of open path-connected sets of $T$

$\blacksquare$