I'm writing this question to briefly inquire how to go about studying Topology based on my experience of having studied a bit of it. I have studied the first half of James Munkres' Topology, except for some units on the countability axioms.
A basic search shows that Topology is the studying of classifying spaces that are homeomorphic, or spaces that are identical to other under (continuous) deformations. I feel as if the the first half of the course on point-set topology was a bit of a let down in this regard. Nowhere did such issues arise; all what we did was defined various toplogical concepts and see the implications and relations between certain toplogical concepts on the toplogy defined on a space.
My questions are:
How is topology pedalogically taught?
After learning point-set topology, when/how can one go about learning toplogy with the focus I mentioned above? Does one have to venture into the study of Algebraic Topology?
On the whole, I am looking for responses that explain how is the subject taught pedalogically and how should one go about studying it?