Path-Connected Components are equal to the Connected Components in a Topological Manifold
# Statement
Let $X$ be a Topological Manifold. Then the collection $\mathcal{P} := {P \subset X : P \text{ is a path-component of }X}$ is equal to $\mathcal{C} := {C \subset X : C \text{ is a connected component of }X}$.
# Proof
Topological Manifolds are Locally Path-Connected and Path-Connected Components are equal to the Connected Components in a Locally Path-Connected Space. $\blacksquare$