# Simply Connected Space **A topological space in which any path between two points can be continuously transformed into any other path between the same two points.** ![[SimplyConnectedSpace.svg|600]] *In a non-simply connected space, the two solid lines between the same endpoints cannot continuously transform into each other without leaving the space, such as the dashed line* > [!NOTE] Simply Connected Space > A topological space $X$ is *simply connected* if for two paths $p:[0, 1]\rightarrow X$ and $q:[0, 1]\rightarrow X$ with the same endpoints, $p$ can be continuously transformed into $q$, that is, there exists a [[homotopy]] between them. > > Equivalently, a topological space $X$ is simply connected if any loop in $X$ can be contracted to a point.