Retracts inherit connectedness and compactness, and are closed inside Hausdorff spaces.Suggest a slogan
Let \(A\subseteq X\) be a retract, with retraction \(r\colon X\to A\). If \(X\) is Hausdorff, then \(A = \theset{x\in X \st r(x) = x}\) is closed in \(X\). If \(X\) is connected or compact, then so is \(A = r(X)\) [@Mun00].