Let \(X\) be a topological space. For \(x, y\in X\), put \(x\sim y\) if there exists a path-connected subspace \(U\subseteq X\) with \(x, y\in U\). The equivalence classes of \(\sim\) are the path components of \(X\).
The relation \(\sim\) is an equivalence relation on \(X\), and \(x\sim y\) if and only if there is a path in \(X\) from \(x\) to \(y\).
[@Mun00].