Path Components are Open iff Union of Open Path-Connected Sets/Path Components are Open implies Space is Union of Open Path-Connected Sets
Jump to navigation
Jump to search
Theorem
Let $T = \struct {S, \tau}$ 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 $T$
$\blacksquare$