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:
- Which approach should I take, as a beginner in this subject who does not want to be overwhelmed by technicalities?
- 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?
- 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!
//doi.org/10.1142/9789812385406_0005Cited by:0
Abstract:
Let A and B be connected subspaces of a topological space X, such that A ∩