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.

Also see

 * Closure of Irreducible Subspace is Irreducible