## (DISCONTINUED) #2: Munkres Problem Set 2.13.4

2.13.4

Problem:

1. If $\{\mathcal{T}_\alpha\}$ is a family of topologies on $X$, show that $\bigcap\mathcal{T}_\alpha$ is a topology on $X$. Is $\bigcup\mathcal{T}_\alpha$ a topology on $X$?
2. Let $\{\mathcal{T}_\alpha\}$ be a family of topologies on $X$. Show that there is a unique smallest topology on $X$ containing all collections $\mathcal{T}_\alpha$, and a unique largest topology contained in all $\mathcal{T}_\alpha$.
3. If $X=\{a,b,c\}$, let $\mathcal{T}_1=\{\emptyset,X,\{a\},\{a,b\}\}$ and $\mathcal{T}_2=\{\emptyset,X,\{a\},\{b,c\}\}$. Find the smallest topology containing $\mathcal{T}_1$ and $\mathcal{T}_2$, and the largest topology contained in $\mathcal{T}_1$ and $\mathcal{T}_2$.

1. $\bigcap\mathcal{T}_\alpha$ is a topology on $X$ due to the fact that each $\mathcal{T}_\alpha$ is a topology on $X$ and if $\{\mathcal{T}_\alpha\}$ is empty, then $\bigcap\mathcal{T}_\alpha$ is the discrete topology on $X$. For $\bigcup\mathcal{T}_\alpha$, let $X=\{a,b,c\}$ and define the topologies $\mathcal{T}_1=\{\emptyset,X,\{a\},\{a,b\}\}$ and $\mathcal{T}_2=\{\emptyset,X,\{a\},\{b,c\}\}$. If we take an intersection of $\mathcal{T}_1$ and $\mathcal{T}_2$, then we are presented with a situation where $\{a,b\}\cap\{b,c\}=\{b\}\notin\mathcal{T}_1\cup\mathcal{T}_2$. Therefore showing that $\bigcup\mathcal{T}_\alpha$ is not a topology on $X$. $\blacksquare$
2. We must note that there exists a topology on $X$ containing all the collections $\mathcal{T}_\alpha$ (discrete topology). Thus, there exists a topology on $X$ contained in all $X$. Therefore denoted as $\mathcal{T}_l=\bigcap\mathcal{T}_\alpha$ to be the largest topology on $X$. For the smallest topology on $X$ that contains all the collections $\mathcal{T}_\alpha$, we take the intersection of all topologies that satisfy the property of containing the union of all the $\mathcal{T}_\alpha$ such that $\mathcal{T}_s=\bigcap_{\bigcup\mathcal{T}_\alpha\subset F}F$ where there exists at least one $F$. Therefore I claim that $\mathcal{T}_l$ is the largest topology and $\mathcal{T}_s$ is the smallest topology on $X$. To show the uniqueness of $\mathcal{T}_s$, let $F$ be a topology that contains all the $\mathcal{T}_\alpha$. $\mathcal{T_s}$ is the intersection of $X$ and therefore $\mathcal{T}_s\subset F$. Then suppose $\mathcal{T}'_s$ fits the criteria of $\mathcal{T}_s$, then $\mathcal{T}'_s\subseteq\mathcal{T}_s$ and $\mathcal{T}_s\subseteq\mathcal{T}'_s$. And therefore $\mathcal{T}_s=\mathcal{T}'_s$, thus $\mathcal{T}_s$ is unique. The uniqueness proof of $\mathcal{T}_l$ is similar. $\blacksquare$
3. The smallest topology containing $\mathcal{T}_1$ and $\mathcal{T}_2$ is the union of $\mathcal{T}_1$ and $\mathcal{T}_2$ such that $\mathcal{T}_1\cup\mathcal{T}_2=\{\emptyset,X,\{a\},\{a,b\},\{b,c\},\{b\}\}$ for which $\{b\}$ has to be thrown in where $\{b\}=\{a,b\}\cap\{b,c\}$. The largest topology contained in $\mathcal{T}_1$ and $\mathcal{T}_2$ is the intersection such that $\mathcal{T}_1\cap\mathcal{T}_2=\{\emptyset,X,a\}$. $\blacksquare$