---- > [!theorem] Theorem. ([[continuity preserves compactness]]) > The image of a [[compact]] [[topological space|space]] under a [[continuous]] function is [[compact]]. > [!proof]- Proof. ([[continuity preserves compactness]]) >Let $X$ and $Y$ be [[topological space]]s and $f:X \to Y$ a [[continuous]] map between them, where $X$ is [[compact]]. We will use [[compactness characterization for subspaces]] (note: this result is often employed implicitly), and also that [[preimages and unions commute]] and [[images and unions commute]]. > Let $\mathscr{A}=\{ A_{\alpha} \}_{\alpha \in J}$ be a [[cover|covering]] of $f(X)$. By [[continuity]], $f^{-1}(\bigcup_{\alpha \in J}^{}A_{\alpha})=\bigcup_{\alpha \in J}^{}f^{-1}(A_{\alpha})$ is an [[cover|open covering]] of $X$. Using compactness, obtain a finite subcover of $X$, $X=\bigcup_{i=1}^{n}f^{-1}(A_{i})$. Then we have $f(\bigcup_{i=1}^{n}f^{-1}(A_{i}))=\bigcup_{i=1}^{n}A_{i} \supset X$ which is a finite subcollection of $\mathscr{A}$ covering $f(X)$. So, $f(X)$ is [[compact]]. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as 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 > ```