Discrete Space is Locally Path-Connected

Theorem

Let $T = \struct {S, \tau}$ be a discrete topological space.

Then $T$ is locally path-connected.


Proof

From Set in Discrete Topology is Clopen, $\set a$ is open in $T$.

From Basis for Discrete Topology, the set:

$\BB := \set {\set x: x \in S}$

is a basis for $T$.


Let $\set x \in \BB$.

From Point is Path-Connected to Itself, it follows that $\set x$ is path-connected.

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

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

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $10$
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