Locally Connected Separable Topological Space has Countably Many Components

Theorem

Let $\struct {X, \tau}$ be a locally connected separable topological space.

Let $\CC$ be the set of components of $\struct {X, \tau}$.

Then $\CC$ is countable.

Proof

From Component of Locally Connected Space is Open:

$\CC \subseteq \tau$

From Equivalence Classes are Disjoint, we have:

the sets in $\CC$ are pairwise disjoint.

From Collection of Pairwise Disjoint Open Sets in Separable Topological Space is Countable, it follows that:

$\CC$ is countable.

$\blacksquare$