Algebraic-Topology/Differential Topology books that also introduce General Topology

Question

Could you recommend me some books in algebraic topology and/or differential topology that introduce enough concepts in point-set topology (compactness, connectedness, continuity, homeomorphism, etc.) that are suitable for a first introduction to point-set topology? It would be good if they cover Urysohn's lemma, Tychonoff's theorem and the Arzelà-Ascoli theorem too.

I am hoping for books that introduce the general topology that is enough for studying algebraic topology and differential geometry.

Answer 1

My book Topology and Groupoids has its first half giving a geometric approach to general topology appropriate for algebraic topology, including adjunction spaces, finite cell complexes, with projective spaces as examples, and function spaces. It does not include the more analysis oriented theorems you mention.

This book's almost unique use in algebraic topology texts of the fundamental groupoid on a set of base points is of course appropriate for discussion of unions of non connected spaces such as the circle, see this mathoverflow discussion, and was supported by Grothendieck in Section 2 of his 1984 Esquisse d'un Programme. Other background to the methodology is in this paper Modelling and Computing Homotopy Types: I.

Answer 2

Bredon's Topology and Geometry is perhaps an option, as it goes swiftly through the requisite general topology in the first chapter, with Tychonoff's Theorem on p. 23 and Urysohn's Lemma on p. 29. (Arzelà-Ascoli is not covered.)

That said, if you have seen no general topology at all before, it might be better to look elsewhere first. Some suggestions are given in the answers to Best book for topology? and there is also Alex Wertheim's recommendation in the comments to try Lee's Introduction to Toplogical Manifolds. As a rough comparison with Bredon of pace/detail, Lee reaches Tychonoff's Theorem on p. 97 and Urysohn's Lemma on p. 112 in the second edition.