Nullhomotopic maps and contractible spaces

Let $f:X\to Y$ be continuous. The map $f$ is nullhomotopic if there is a point $y_0\in Y$ such that $f$ is homotopic to the constant map $c_{y_0}:X\to Y$, $c_{y_0}(x)=y_0$ (Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints).

A nonempty topological space $X$ is contractible if every continuous map $f:X\to Y$ to every topological space $Y$ is nullhomotopic.

This definition separates the property of the space from the particular map $\operatorname{id}_X$. The next corollary proves that it is equivalent to the familiar condition that the identity map be nullhomotopic.

• Nonemptiness is included so that a contracting point can be named. The empty space is not called contractible under this convention.
• The constant to which a map is homotopic may depend on the map. Contractibility does not assert that two arbitrary constant maps into a disconnected target are homotopic.