Definition:Path Component/Union of Path-Connected Sets

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Definition

Let $T$ be a topological space.


The path component of $T$ containing $x$ is defined as:

$\displaystyle \map {\operatorname{PC}_x} T = \bigcup \left\{{A \subseteq S: x \in A \land A}\right.$ is path-connected $\left.\right\}$