I'm looking for a collection of results in basic topology. Often I find myself working out some small detail for i.e. continuous functions, and I know I have done the same thing before, but cannot remember if the result was $=$ or just $\subseteq$ or what exact conditions the map has to have for the result to hold.
A small example: If $f\colon X\rightarrow Y$ is a continuous map and $A\subseteq Y$, then in general only $f^{-1}(\overline{A})\subseteq \overline{f^{-1}(A)}$. But if $f$ is also an open map, then $f^{-1}(\overline{A})=\overline{f^{-1}(A)}$.
Do you know a good (not necessarily small) collection of results? Most topology books cover some very elementary ones, but I never find what I'm looking for, or when I eventually find something, it would have been faster if I had actually proven it myself.
Thank you in advance.
Best Answer
When I took real analysis and topology (undergrad) I used this wikibook for proofs.
It has definitions, theorems (sometimes with proofs), and exercises.
If you do not like your book, try to find the one you like in the library, at least that is what I did.