# Indiscrete Space is Path-Connected

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## Theorem

Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ be an indiscrete topological space.

Then $T$ is path-connected.

## Proof

Let $a, b \in S$.

Consider any mapping $f: \left[{0 \,.\,.\, 1}\right] \to S$ such that $f \left({0}\right) = a$ and $f \left({1}\right) = b$.

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

The result follows by definition of path-connectedness.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 4: \ 9$