Connected Component is Closed

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Theorem

Let $T = \struct{S, \tau}$ be a topological space.


Then every connected component of $T$ is closed.

Proof

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

By the definition of connected component, $H$ is connected.

From Closure of Connected Set is Connected then the closure $H^-$ is connected.

By the definition of the closure, $H \subseteq H^-$.

By the definition of connected component, $H$ is a maximal connected set.

Hence $H = H^-$.

From Set is Closed iff Equals Topological Closure, $H$ is closed.

$\blacksquare$

Sources