# Closure of Connected Set is Connected

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## Theorem

Let $T$ be a topological space.

Let $H$ be a connected set of $T$.

Let $H^-$ denote the closure of $H$ in $T$.

Then $H^-$ is connected in $T$.

## Proof

By Set is Subset of Itself, the result follows by setting $K = H^-$ in Set between Connected Set and Closure is Connected.

$\blacksquare$

## Also see

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness