Path component

Definition

Let $X$ be a topological space. For $x, y \in X$, write $x \simeq y$ when there is a path in $X$ from $x$ to $y$. This is an equivalence relation on $X$. Its classes are the path components of $X$.

Remarks

The relation is reflexive by constant paths, symmetric by $t \mapsto \gamma(1 - t)$, and transitive by joining two paths one after the other. The path component of $x$ is the largest path-connected subset of $X$ that contains $x$. Each path component lies in a connected component. The topologist's sine curve is one connected component but has two path components. Path components need not be closed.

Explanations

L1 human

The path component of a point is the set of all points that you can reach from it along a path.