Topological retraction/Definition

Definition

Let $(X,\mathcal{ J })$ be a topological space and let $A\in\mathcal{P}(X)$ be considered a s subspace of $X$. A continuous map, $r:X\rightarrow A$ is called a retraction if:

• The restriction of $r$ to $A$ (the map $r\vert_A:A\rightarrow A$ given by $r\vert_A:a\mapsto r(a)$) is the identity map, $\text{Id}_A:A\rightarrow A$ given by $\text{Id}_A:a\mapsto a$

If there is such a retraction, we say that: $A$ is a retract of $X$.