## (DISCONTINUED) #3: Munkres Problem Set 2.13.5

2.13.5

Problem: Show that if $\mathcal{A}$ is a basis for a topology on $X$, then the topology generated by $\mathcal{A}$ equals the intersection of all topologies on $X$ that contain $\mathcal{A}$. Prove the same if $\mathcal{A}$ is a subbasis.

Answer: Let $\{\mathcal{T}_\alpha\}$ be a family of all topologies on $X$ that contains the basis $\mathcal{A}$ and $\mathcal{T}_\mathcal{A}$ be the topology generated by the basis $\mathcal{A}$. Note $\bigcap T_\alpha$ is a topology on $X$ and $\bigcap T_\alpha\subset T_\mathcal{A}$ since $\mathcal{A}\subset\mathcal{T}_\mathcal{A}$. On the other hand, any $U\in\mathcal{T}_\mathcal{A}$ is a union of elements in $\mathcal{A}$, so $U\in T_\alpha$ for all $\alpha$ and thus $\mathcal{T}_\mathcal{A}\subset\bigcap T_\alpha$. $\blacksquare$

Happy February! Sorry for the short Mondays. The time is ticking for my research paper and I need to have the abstract and the main portions of my paper finished in exactly three weeks. I also do not want to just skim the next section (2.13.5 is the last problem of Section $13$) and then try to pull out dumb responses to the problems in the text. I will be leaving Section $16$ problems for next Monday where I will be doing at least two to three problems to make up for my “lesser” days. There will also be another interesting discussion this Thursday so look out for that!