Closure of Connected Set is Connected

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

• Closure of Irreducible Subspace is Irreducible

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