Definition 4

Let X be a topological space and let A be a subset of X.

A retraction of X onto A is called a continuous mapping from X into A which is a retraction of the canonical injection of A into X. If there exists a retraction of X onto A, we say that X can be retracted onto A, or also that A is a retract of X.

Let X be a topological space and let A be a subspace of X.

Suppose that there exists a retraction $r$ of X onto A. The subspace A is identified with the set of points $x\in X$ such that $x=r(x)$. If X is a separated space, A is then closed in X.

According to the following lemma, for a subspace A to be a retract of X, it is necessary and sufficient that every continuous mapping from A into a topological space Y extend to a continuous mapping from X into Y.