> [!definition] > > Let $X$ be a [[Topological Space|topological space]], then the following are equivalent: > 1. $X$ is [[Homotopy Equivalence|homotopy equivalent]] to a point equipped with the trivial topology. > 2. $\text{Id}_X$ is [[Homotopy|homotopic]] to a constant function $g: X \to X$. > > If the above holds, then $X$ is **contractible**. > [!theorem] > > Let $X$ be a contractible space, then $X$ is [[Path-Connected|path-connected]].