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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.

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The resource you are searching for: it will be 10 times better than any one you find on the internet if you write it for yourself :) I'm not trying to be sarcastic, I'm just saying that it's well worth the time investment. Specifically the organization and content will be ideal for you. –  rschwieb Oct 19 '12 at 20:42
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I find keeping a list of definitions much more valuable. Re-deriving results until they become instinctual is actually a good thing - it means you are increasing your understanding of the subject. (I don't this is necessarily true for all subjects, but general topology is a case where almost all the basic results follow straight from the definitions.) –  Thomas Andrews Oct 19 '12 at 20:47
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up vote 2 down vote accepted

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.

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