> [!theorem] > > Let $M$ be an $E$-[[Manifold|manifold]]. If $E$ is [[Locally Convex Topological Vector Space|locally convex]], then $E$ is [[Locally Path-Connected|locally path-connected]], and > 1. $M$ is locally path-connected. > 2. The [[Connected Component|connected components]] of $M$ are the same as its path components. > 3. $M$ is connected if and only if it is path-connected. > 4. If $M$ is [[Second Countable|second countable]], then $M$ has countably many [[Connected Component|connected components]]. > > *Proof*. Since every convex set is path connected, $E$ is locally path-connected. The next two properties follows from properties of locally path-connected spaces. > > Lastly, if $M$ is second countable, then $M$ is separable. Let $F \subset M$ be a countable [[Dense|dense]] subset, then since each path component is open, the path components of each point in $F$ covers $M$. Therefore $M$ can only have countably many connected components.