Proposition 5

In a topological space, the relation “there exists a path with origin $x$ and term $y$” is an equivalence relation.

The path-connected components of a topological space X are called the equivalence classes for the above relation. A path-connected component is a path-connected space. Let $x$ be a point of X. The path-connected component of $x$ is the union of the path-connected subspaces of X containing $x$. It is also the union of the images of the paths with origin $x$ in X.