📂Topology

Path Connectivity Components

Definition ¹

A path connected component of a topological space $X$ is a path connected subset that has itself as the only connecting superSet. Specifically, the path connected component that includes $x \in X$ is written as $P_{x}$.

Theorem

• [1]: $x \in X$ belongs to only one $P_{x}$.
• [2]: For $a,b \in X$, it is either $P_{a} = P_{b}$ or $P_{a} \cap P_{b} = \emptyset$.
• [3]: Every path connected space is a subset of some path connected component.
• [5]: $X$ being a path connected space is equivalent to $X$ having only one path connected component.

Difference from Connected Components

At first glance, there seems to be no difference from connected components, but a closer look reveals that theorem [4] is subtly missing. Its property is as follows.

• [4]: All the connected components of $X$ are closed sets in $X$.

When changing ‘connected’ to ‘path connected’ in [4], a counterexample is the topologist’s sine curve.