Problem: Show that if is a basis for a topology on , then the topology generated by equals the intersection of all topologies on that contain . Prove the same if is a subbasis.
Answer: Let be a family of all topologies on that contains the basis and be the topology generated by the basis . Note is a topology on and since . On the other hand, any is a union of elements in , so for all and thus .
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 ) and then try to pull out dumb responses to the problems in the text. I will be leaving Section 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!