(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!

 

 

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