Discrete Space is Locally Path-Connected

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Theorem

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

Then $T$ is locally path-connected.


Proof

From Set in Discrete Topology is Clopen, $\left\{{a}\right\}$ is open in $T$.

From Basis for Discrete Topology, the set:

$\mathcal B := \left\{{\left\{{x}\right\}: x \in S}\right\}$

is a basis for $T$.


Let $\left\{{x}\right\} \in \mathcal B$.

From Point is Path-Connected to Itself, it follows that $\left\{{x}\right\}$ is path-connected.

Hence $T$ has a basis consisting entirely of path-connected sets.

So by definition $T$ is locally path-connected.

$\blacksquare$


Sources