By assumption there exists a retraction \(r:X \to A\) which is homotopic to \(\mathbbm{1}_{X}\). Let \(\iota:A \to X\) be the inclusion map which is continuous since topology on \(A\) is subspace topology. Since \(r = \iota r\), \(\iota r\) is homotopic to \(\mathbbm{1}_{X}\). The map \(r \iota:A \to A\) is the identity map since \(r \restrict{A} = \mathbbm{1}_{A}\) hence \(r \iota\) is homotopic to \(\mathbbm{1}_{A}\). Thus, \(X \simeq A\).