How can one get better at topology?

Question

Some users on this site are insanely good at topology. For example: Eric Wofsey and Brian M. Scott. There are a few others as well.

How did they get so good? What's the secret? What tips can they give?

I'm asking because I'm working through Munkres right now and even though I think I'm doing well, there's some questions which I get utterly stuck on, and I look them up on this site. The answers to some of these questions are very slick, almost ingenious, which makes me wonder how one would come up with them.

This might not be the best question for this site. Nevertheless, I think getting some insight into how the minds of the best mathematicians work is relevant to a lot of people.

Answer 1

I am not an expert in topology, but studying counter examples helped me a lot. For example, when a theorem says:

If a closed and a compact subset of $\mathbb R^n$ are disjoint, then there is a positive distance between them.

Although studying the proof of the above theorem is important, I think you need to ask yourself:

"what if both sets are only closed but neither is compact?"

I think this way you can get rid of wrong assumptions about topology. For example, you can start with:

What is true in Hausdorff spaces but not necessarily true otherwise?

Answer 2

I bet their secret spell is hard work!

You might consider to look up some of the books suggested in the post "Best book for topology?" and work through as many problems as possible.

Also, whenever reading mathematics, it is a good thing to try to understand the theorems, and, if possible, find your own proof before reading the particular proof given.