Problem: Let $X$ be a topological space and $\{ \mathcal{T} _\alpha \} $ be a family of topologies on $X$.
-
Show that $\bigcap_{\alpha } \mathcal{T} _\alpha $ is a topology on $X$. Is $\bigcap_{\alpha } \mathcal{T} _\alpha$ a topology on $X$?
-
Show that there is a unique smallest topology on $X$ containing each of the topologies $\mathcal{T} _\alpha $ and a unique largest topology contained in each of the topologies $\mathcal{T} _\alpha $.
-
If $X = \{ a,b,c \} $, let
\[
\mathcal{T} _1 = \{ \emptyset , X, \{ a \}, \{ a \} \} \quad \text{ and } \quad \mathcal{T} _2 = \{ \emptyset , X, \{ a \} , \{ b,c \} \} .
\]
Find the smallest topology containing $\mathcal{T} _1$ and $\mathcal{T} _2$, and the largest topology contained in $\mathcal{T} _1$ and $\mathcal{T} _2$.
Solution: I encourage you to attempt to solve the problem today. The solution will be provided tomorrow. This will give you the opportunity to test your understanding of the problem and to improve your skills in solving similar problems in the future.