Path Component

Definition

Consider a Topological Space $(X,\mathcal{T})$. The path component $P_x$ of a point $x\in X$ is the union of all path connected subsets of $X$ that contains $x$. It is the unique largest (with respect to $\subseteq$) path connected subset of $X$ that contains $x$.

Facts

Each point $x\in X$ is contained in exactly one path component.

Given points $x,y\in X$, their path components $P_x,P_y$ are the same or disjoint. $\forall x,y\in X,P_x=P_y\text{or}{P}_x\cap P_y=\varnothing$

Every path connected subset in $X$ is contained in some path component.

$X$ is a Path Connected Space if and only if $X$ consists of a single component.