Let $(X,\tau)$ be a [topological space](/page/Topological%20Space). For each $x \in X$, let $P_x$ denote the path component of $x$, equivalently the equivalence class of $x$ under the relation $x \sim_p y$ if and only if there exists a path in $X$ from $x$ to $y$. Then the path components of $X$ are pairwise disjoint path-connected subspaces of $X$, and

\begin{align*} X = \bigcup_{x \in X} P_x. \end{align*}

Moreover, if $A \subset X$ is nonempty and path-connected with the [subspace topology](/page/Subspace%20Topology), then there exists $x \in X$ such that $A \subset P_x$.