----- Let $X$ be a set; let $\mathscr{B}$ be a [[topology generated by a basis|basis for a topology]] $\tau$ on $X$. > [!proposition] Proposition. ([[open sets are unions of basis elements]]) > Then $\tau$ equals the collection of all unions of elements of $\mathscr{B}$. > [!proof]- Proof. ([[open sets are unions of basis elements]]) > We'll show a two-way inclusion. > 1. Since $\mathscr{B} \subset \tau$, any union of elements of $\mathscr{B}$ is in $\tau$ by definition of [[topological space]]. > 2. Consider $U \in \tau$. For each $x \in U$ we can find $B_{x} \in \mathscr{B}$ such that $x \in B_{x} \subset U$. Then $U=\bigcup_{x \in U} B_{x}$ and so we've written each element $U$ of $\tau$ as a union of basis elements. $\textcolor{Apricot}{\text{Warning:}}$ This expression for $U$ is not unique! ----- #### ---- #### 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 > ```