----- > [!proposition] Proposition. ([[when closed sets behave like open sets]]) > Let $X$ be a [[topological space]]. Then the following hold: > 1. $\emptyset, X$ are [[closed set|closed in]] $X$; > 2. (*Finite* unions of closed sets are closed) The *finite* union of [[closed set]]s in $X$ is again [[closed set|closed in]] $X$; >3. (*Arbitrary* intersections of closed sets are closed) The intersection of [[closed set]]s in $X$ is again [[closed set|closed in]] $X$. > [!proof]- Proof. ([[when closed sets behave like open sets]]) > >1. $X^c = \emptyset$ and $\emptyset^c = X$. >2. Given a collection of [[closed set]]s $\{ A_\alpha \}_{\alpha \in J}$, we apply DeMorgan's law: $X - \bigcap_{\alpha \in J}^{} A_{\alpha} = \bigcup_{\alpha \in J}^{} (X - A_{\alpha}).$ >3. Given a finite collection of [[closed set]]s $A_{1},\dots,A_{n}$, we again apply DeMorgan's law: $X - (A_{1} \cup \dots \cup A_{n})= \bigcup_{i=1}^{n} (X-A_{i}).$ ----- #### ---- #### 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 > ```