Reddit reviews Counterexamples in Topology;Dover Books on Mathematics

This reminds me of Counterexamples in Topology which is literally a book of pathological shapes and sets that frequently serve as counterexamples.

There is an excellent series of Counterexamples in ... books which might be relevant to this thread:

counterexamples in...

• calculus
  
• analysis
  
• topology
  
• probability & statistics
  
• several complex variables
  
  Or just run a search on amazon for "Counterexamples in".

So many of the counterexamples in here or here (no link to the 2nd).

I think the worst for me is that some convergent series can be rearranged to be divergent. Fucking conditional convergence...

If you love calculus it might be worth brushing up on your real and complex analysis. It's the pure maths side of calculus (or at least can usefully be regarded as such. That's not a 100% accurate description). I'm not sure what to offer for that that isn't very clearly in the class of a textbook and expensive though. :-) Tom Korner's "A Companion to Analysis" is a good intro though.

One book which I really enjoyed is (don't laugh) Schaum's Outlines on Advanced Calculus. It's got a lot of really good exercises and teaches you some unconventional approaches to solving integrals.

I'll also second the recommendations for "proofs from the book" and "proofs and refutations"

One little maths book which I loved (and you have to be slightly odd kind of person to say that about it) is Counter examples in topology. It's an entire book devoted to perverse examples. I really enjoy the thought process that goes into such things. But if you don't know any topology that's not a very good recommendation. :-)

Counterexamples in topology

Counterexamples in analysis