Let $(X,\tau)$ be a [topological space](/page/Topological%20Space). Define two points $x,y\in X$ to be path equivalent if there exists a continuous map $\gamma:[0,1]\to X$, where $[0,1]$ has the [subspace topology](/page/Subspace%20Topology) inherited from $\mathbb{R}$, such that $\gamma(0)=x$ and $\gamma(1)=y$. The equivalence classes for this relation, called the path components of $X$, form a partition of $X$. Each path component is path connected. Moreover, if $A\subset X$ is path connected, then for every $a\in A$, the set $A$ is contained in the path component of $X$ containing $a$.