How should I deal with point-set topology problems in algebraic topology?

Question

In algebraic topology, many point-set topology problems arise. For example, the product of quotient maps need not be a quotient map; the smash product may not be associative; the canonical map $Z^{X\times Y}\cong(Z^Y)^X$ is not necessarily a homeomorphism...

It seems that many authors chooses to stay in the category of compactly generated (weak Hausdorff) spaces to remedy this. For example, in Chapter 5 of Algebraic Topology by J. P. May, he introduces this concept and states without proof some basic properties, and then he assumes all topological spaces are compactly generated in the remainder of the book.

On the other hand, some authors do not assume this. Then some restrictions are necessary. In this context local compactness frequently crops up.

Here are my questions:

1. Which approach should I take, as a beginner in this subject who does not want to be overwhelmed by technicalities?
2. I currently know nothing about compactly generated spaces. I really want to read the text by May, but he uses this throughout, making many of his assertions simply false for me without the necessary restrictions (for instance, a cofibration need not be closed). How should I deal with this problem?
3. Are there any readable introductions (be it a book, an article, lectures notes, etc.) to compactly generated spaces that provides working knowledge for use in algebraic topology?

Thanks for any advice!

Answer 1

Some thoughts:

1. You should learn to cope with different assumptions. This will take time; I'm sorry. You will not learn all this overnight. You will commonly come across CW restrictions, simplicial stuff, and the compactly-generated category. There is really no way around this, since all these categories have their own intrinsic advantages. Ease with these notions comes only with experience using them.

2. See point 1. When you need to, step aside and prove what he says is obvious. Or, just assume it's all fine and keep reading. Sweat the technicalities when you need to, not when someone else does.

3. Steenrod's paper A Convenient Category of Topological Spaces was one of the first to advocate for compactly-generated spaces. It actually contains proofs of many of the things you're asking about.

Michigan Math. J., Volume 14, Issue 2 (1967), 133-152.