As an introductory remark for this being my first set of problems from the Munkres text, I have a few things to note.
- The numbering system corresponds as follows: “<chapter>.<section>.<problem number>”.
- Please feel free to correct any mistakes that I make in the comments and I will update them, as I am doing this for my benefit and the readers benefit as well.
- The schedule for these posts will solidify for every Monday at AM PST.
- You can find Munkres’s Topology in the References section on my blog if you need a quick link to the text at your convenience.
For this problem set, I am sorry it has only two problems. I will try to post a few more following up in the middle of the week so that can everyone see more. If not, we will start fresh next week.
Problem: Let be a topological space; let be a subset of . Suppose that for each there is an open set containing such that . Show that is open in .
Answer: Let denote the open set where . For each , there will be an open set associated with it. Therefore . By the definition of an open set, is an open set by the fact that the union of opens sets forms an open set.
Problem: Show that the collection given in Example of Section is a topology on set . Is the collection
is infinite or empty or all of
a topology on ?
Answer: By the definition of a topology,
- and ; therefore and are in .
- Let be an indexed family on nonempty elements of . ThenTherefore this shows that for all .
- And Therefore this shows that for some .
Thus is a topology on .