# Indiscrete Space is Path-Connected

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## Theorem

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.

Then $T$ is path-connected.

## Proof

Let $a, b \in S$.

Consider any mapping $f: \closedint 0 1 \to S$ such that $\map f 0 = a$ and $\map f 1 = b$.

From Mapping to Indiscrete Space is Continuous, we have that $f$ is continuous.

The result follows by definition of path-connectedness.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $4$. Indiscrete Topology: $9$