I was tyring to read this fact that the connected sum of two disks is homeomorphic to annulus. By intutitive picture, it is obvious, but I wanted to it in a formalistic way. So I was reading many different textbooks, looking up the formal definition of connected sum, surfaces, etc..
The definition of connected sum includes the notion of homeomorphism. So when they prove the above statement formally, do they use various kinds of homeomorphism, and actually constrct a homeomorphism between the connected sum and annulus?
Anyway in general, if I want to study these things in a fully formalized context, what topics should I study? And could you refer me any rigorous textbook?