----- > [!proposition] Proposition. ([[products preserve deformation retracts]]) > Let $X,Y$ be [[topological space|topological spaces]] and $A \subset Y$ a [[deformation retract]]. Prove $X \times A$ is a [[deformation retract]] of $X \times Y$. > [!proof]- Proof. ([[products preserve deformation retracts]]) > > Suppose $A$ is a [[deformation retract]] of $Y$ via a [[homotopy]] $H: Y \times I \to A$ which satisfies > - $H(y,0)=y$ for all $y \in Y$ > - $H(y,1) \in A$ for all $y \in Y$ > - $H(a,t) \in A$ for all $a \in A$ and $t \in I$. > > If we define $F:X \times Y \times I \to X \times A$ by $F\big( (x,y), t \big):= \big( x, H(y,t) \big)$ > then we obtained the desired properties, in particular this $F:X \times Y \times I \to X \times A$ satisfies > - $F\big( (x,y), 0 \big)=(x,y)$ for all $(x,y) \in X \times Y$ > - $F\big( (x,y), 1 \big) \in X \times A$ for all $(x,y) \in X \times Y$ > - $F\big( (x, a), t \big) \in X \times A$ for all $(x,a) \in X \times A$ and $t \in I$ > as required. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch04