Locally Path-Connected Space is Locally Connected

Theorem

Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected.


Then $T$ is also locally connected.


Proof

Let $x \in S$ be any point of $T$.

Let $\BB$ be a local basis of path-connected sets for $x$.

From Path-Connected Space is Connected, $\BB$ is a local basis of connected sets.

Thus, $T$ is locally connected by definition.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness
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