A topological space \(X\) is contractible if \(\id_X\) is nullhomotopic: there is a point \(x_0\in X\) such that \(\id_X\) is homotopic to the constant map \(c\colon X\to X\), \(c(x) = x_0\).
A topological space \(X\) is contractible if and only if \(X\) is homotopy equivalent to a one-point space \(\ts{x_0}\), that is, there are continuous maps \(f\colon X\to\ts{x_0}\) and \(g\colon\ts{x_0}\to X\) with \(f\circ g = \id_{\ts{x_0}}\) and \(g\circ f\simeq\id_X\).
See [@Hat02].