Chapter 2.13, Problem 4E

(a) If $\left\{T_{\alpha }\right\}$ is a family of topologies on $X$, show that ${\displaystyle \cap T_{\alpha }}$ is a topology on $X$. Is ${\displaystyle \cup T_{\alpha }}$ a topology on $X$?

(b) Let $\left\{T_{\alpha }\right\}$ be a family of topologies on $X$. Show that there is a unique smallest topology on $X$ containing all the collections $T_{\alpha }$, and a unique largest topology contained in all $T_{\alpha }$.

(c) If $X=\left\{a,b,c\right\}$, let

$T_1=\left\{\varnothing ,X,\left\{a\right\},\left\{a,b\right\}\right\}$ and $T_2=\left\{\varnothing ,X,\left\{a\right\},\left\{b,c\right\}\right\}$.

Find the smallest topology containing $T_1$ and $T_2$, and the largest topology contained in $T_1$ and $T_2$.