----- > [!proposition] Proposition. ([[topology from closed sets]]) > Let $\mathcal{C}$ be a collection of subsets of the set $X$. Suppose that $\emptyset$ and $X$ are in $\mathcal{C}$, and that finite unions and arbitrary intersections of elements in $\mathcal{C}$ are again in $\mathcal{C}$. Then the collection $\tau=\{ X-C : C \in \mathcal{C} \}$ > is a [[topological space|topology on]] $X$. > [!proof]- Proof. ([[topology from closed sets]]) > 1. $\emptyset$ and $X$ are in $\tau$ by hypothesis. > 2. For an arbitrary union of elements in $\tau$ indexed by $\alpha$, $\bigcup_{\alpha}^{} (X - C_\alpha),$ > we have $\bigcup_{\alpha}^{} (X - C_{\alpha}) = X - \bigcap_{\alpha}^{} C_{\alpha}$ > 3. For a finite intersection $(X - C_{1}) \cap \dots \cap (X-C_{n}),$ > we have $\begin{align} > (X - C_{1}) \cap \dots \cap (X-C_{n}) = & X - \bigcap_{i=1}^{n} C_{i}, > \end{align}$ > which is in $\tau$ since the $C_{i}$ are closed under arbitrary unions. ----- #### ---- #### 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 > ```