The intersection of topologies on a fixed set is again a topology

Let $(T 𝛼) 𝛼 \in 𝐼$ be a family of topologies on a set $𝑋$ . Then the union $T = ⋂ 𝛼 \in 𝐼 T 𝛼$ is again a topology on $𝑋$ . #m/thm/topology

Proof

Since ${\emptyset, 𝑋} \subseteq T 𝛼$ for all $𝛼 \in 𝐼$ , it follows that ${\emptyset, 𝑋} \subseteq T$ . Let ${𝑈 𝛽} 𝛽 \in 𝐽 \subseteq T$ be a family of open subsets under $T$ . Then the union ${𝑈 𝛽} 𝛽 \in 𝐽 \subseteq T 𝛼$ and hence $⋃ 𝛽 \in 𝐽 𝑈 𝛽 \in T 𝛼$ for all $𝛼 \in 𝐼$ , wherefore $⋃ 𝛽 \in 𝐽 𝑈 𝛽 \in T$ . Similarly let ${𝑉 𝑖} 𝑛 𝑖 = 1 \subseteq T$ be a finite family of open subsets under $T$ . Then the intersection ${𝑉 𝑖} 𝑛 𝑖 = 1 \subseteq T 𝛼$ and hence $⋃ 𝑛 𝑖 = 1 𝑈 𝑛 \in T 𝛼$ for all $𝛼 \in 𝐼$ , wherefore $⋃ 𝑛 𝑖 = 1 𝑈 𝑛 \in T$ . Therefore $T$ is a topology on $𝑋$ .

#state/tidy | #lang/en | #SemBr