----- > [!proposition] Proposition. ([[retractions preserve contractibility]]) > Any [[retract]] $A$ of a [[contractible]] [[topological space]] $X$ is [[contractible]]. ^proposition > [!proof]- Proof. ([[retractions preserve contractibility]]) > Using that $X$ is [[contractible]], obtain a constant map $c:X \to X$ such that $c$ is [[homotopy|homotopic]] to $\id_{X}$. $X \xrightarrow{c, \id_{X}} X \xrightarrow{r}_{\supset } A$ Then from [[continuous functions respect homotopy]] it follows that the constant map $r \circ c: X \to A$ is [[homotopy|homotopic]] to $r \circ \id_{X}: X \to A$. Since $r$ is a [[retract|retraction]], it follows that the constant map $(r\circ c)|_{A}$ is [[homotopy|homotopic]] to $(r \circ \id_{X}) |_{A}=r |_{A}=\id_{A}$. Hence $A$ is [[contractible]]. \ \ > ![[continuous functions respect homotopy]] ----- #### ---- #### 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 > ``` #reformatrevisebatch03