Deformation Retract Subspaces of a Topological Space

Recall from the Retract Subspaces of a Topological Space page that if $X$ is a topological space and $A \subset X$ is a topological subspace then $A$ is said to be a retract of $X$ if there exists a continuous function $r : X \to A$ such that $r \circ \mathrm{in} : \mathrm{id}_A$ where $\mathrm{in} : A \to X$ is the inclusion map.

We will now define another type of retract called a deformation retract.

Definition: Let $X$ be a topological space and let $A \subset X$ be a topological subspace. Then $A$ is said to be a Deformation Retract of $X$ if there exists a continuous function $r : X \to A$ if $r \circ \mathrm{in} = \mathrm{id}_A$ and $\mathrm{in} \circ r = \mathrm{id}_X$.

By the definition above, every deformation retract is a retract.

For the first example, consider the space $A = S^1$ which is the unit circle and $X = S^1 \times [0, 1]$ which is the unit cylinder. Then $A$ is a deformation retract of $X$:

For another example, consider the space $A = \{ p \}$ and $X = \{ p, q \}$ where $p, q \in \mathbb{R}^2$ and $p \neq q$ and equip $X$ with the discrete topology. Then $A$ is NOT a deformation retract of $X$ since $A$ is connected and $X$ is disconnected.