# Indiscrete Space is Ultraconnected

Jump to navigation
Jump to search

## Theorem

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

Then $T$ is ultraconnected.

## Proof

There is only one non-empty closed set in $T$.

So there can be no two closed sets in $T$ which are disjoint.

Hence (trivially) $T$ is ultraconnected.

$\blacksquare$

## Sources

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