> [!proposition]+ Proposition. ([[continuous functions respect homotopy]]) > Let $X$, $Y$, $Z$ be [[topological space|topological spaces]]. If $f:X \to Y$ is [[homotopy|homotopic]] to $g:X \to Y$ via the map $H$ and $k: Y \to Z$ is [[continuous]], then $k \circ f$ and $k \circ g$ are [[homotopy|homotopic]] via the map $k \circ H$. > > Compare to [[continuous functions respect path homotopy]]. ^proposition > [!proof]+ Proof. ([[continuous functions respect homotopy]]) > $k \circ H$ is [[continuous]] as a composition of [[continuous]] functions. And $(k \circ H)(x, 0)=(k \circ f)(x) \text{ and } (k \circ H)(x,1)=(k \circ g)(x).$ ^proof #### 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