Reference for general-topology
Solution 1:
For general topology, it is hard to beat Ryszard Engelking's "General Topology". It starts at the very basics, but goes through quite advanced topics. It may be perhaps a bit dated, but it is still the standard reference in general topology.
Solution 2:
Stephen Willard, General Topology
This book is less complete than Engelking, but still contains enough material to make a good reference book. It is also quite cheap, as a Dover book.