----- > [!proposition] Proposition. ([[closed map lemma]]) > If $f:X \to Y$ is a [[continuous]] map between a [[compact]] [[topological space]] $X$ and a [[Hausdorff space|Hausdorff]] [[topological space]] $Y$, then $f$ is closed. \ Immediate is that if $f$ is furthermore a [[bijection]], then it is in fact a [[homeomorphism]]. > [!proof]- Proof. ([[closed map lemma]]) > We shall prove that the images of closed sets of $X$ under $f$ are [[closed set|closed in]] $Y$; this will prove continuity of the map $f^{-1}$. If $A$ is closed in $X$, then by [[closed subspaces of compact spaces are compact]] it is [[compact]], and then by [[continuity preserves compactness]] we get that $f(A)$ is [[compact]]. By [[every compact subspace of a Hausdorff space is closed]] we see that $f(A)$ is closed in $Y$, 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 > ```