Discrete Space is Locally Path-Connected

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space where $\tau$ is the discrete topology on $S$.

Then $T$ is locally path-connected.

Hence $T$ is also locally connected.

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

Then 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.

From Locally Path-Connected Space is Locally Connected, it follows that $T$ is locally connected.