\begin{definition}[Definition:Path Component/Union of Path-Connected Sets]
Let $T$ be a topological space.
The '''path component of $T$ containing $x$''' is defined as:
:$\ds \map {\operatorname{PC}_x} T = \bigcup \leftset {A \subseteq S: x \in A \land A}$ is path-connected $\rightset {}$
Category:Definitions/Path-Connected Spaces
\end{definition}