Topological Manifolds have Countably many Connected Components

Last updated Nov 1, 2022

Let $M$ be a Topological Manifold of Manifold Dimension $n$. $M$ has countably many Connected Components.

Because

• Topological Manifolds are Locally Path-Connected
• Connected Components of Locally Path-Connected Spaces are Open
• Connected Components Partition the Space

we have that our Connected Components form an Open Cover of $M$, which we denote as $\mathcal{C}$. Because $M$ is Second Countable by definition and Open Covers in a Second Countable space reduce to a Countable Subcover, we can find an Open Subcover of Connected Components $\mathcal{C}’ \subset \mathcal{C}$. Since $\mathcal{C}$ is a Partition of $M$, if $\mathcal{C}’ \subsetneq \mathcal{C}$, there would be some partition set of $\mathcal{C}$ not included and $\mathcal{C}’$, but then $\mathcal{C}’$ would not be an Open Cover. Thus $\mathcal{C}’ = \mathcal{C}$. $\blacksquare$