----- > [!proposition] Proposition. ([[closed subspaces of compact spaces are compact]]) > Every [[closed set|closed]] [[subspace topology|subspace]] of a [[compact]] space is [[compact]]. > [!proof]- Proof. ([[closed subspaces of compact spaces are compact]]) > Let $Y$ be a closed subspace of the [[compact]] space $X$. We'll use the [[compactness characterization for subspaces]]. Given an [[cover|open covering]] $\mathscr{A}$ of $Y$ by sets open *in $X$*, let us form an open covering of $X$ by adjoining to $\mathscr{A}$ the single open [^1] set $X-Y$, that is $\mathscr{B}=\mathscr{A} \cup \{ X-Y \}.$ Now, using [[compact]]ness of $X$ obtain an finite subcollection of $\mathscr{B}$ covering $X$. If this subcollection contains the set $X-Y$, discard $X-Y$, otherwise leave this subcollection alone. The resulting collection is a finite subcollection of $\mathscr{A}$ that covers $Y$. [^1]: This is where needed that $Y$ is closed in $X$. ----- #### ---- #### 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 > ```