12:42 PM
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I have no doubt that you there are many useful online resources for these, but many such results are available here at MSE, together with their proofs. I'll give a list of some basic results about images and preimages links to the posts, which have proofs here at MSE. I am making this CW, so fee...

Is there a similar list for stronger results when the map is continuous? — plebmatician 30 mins ago
I am not sure what the user who posted the above comment means - for example, I cannot imagine something I can specifically say about $f[A\cap B]$ if I have the additional assumptions that the function is continuous.
Perhaps properties like this are mean in that comment: preimage of an open set (under continuous function) is open, continuous image of a compact set is compact, continuous image of a compact set is compact.
I am posting this here in case somebody might have an idea what is actually meant here.
@plebmatician To be honest, I am not really sure what kind of properties you have in mind. In any case, I have mentioned this in Calculus and analysis chatroom, maybe somebody will notice it there. — Martin Sleziak 11 secs ago