# Quasicomponents and Path Components are Equal in Locally Path-Connected Space

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## Theorem

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

Then $A \subseteq S$ is a path component of $T$ if and only if $A \subseteq S$ is a quasicomponent of $T$.

## Proof

## 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