Kelley J.L. General Topology

Mineola, New York : Dover Publications, Inc., 2017. — 320 p. — (Dover Books on Mathematics). — ISBN: 9780486815442.This book is a systematic exposition of the part of general topology which has proven useful in several branches of mathematics. It is especially intended as background for modern analysis, and I have, with difficulty, been prevented by my friends from labeling it: What Every Young Analyst Should Know.
The book, which is based on various lectures given at the University of Chicago in 1946-47, the University of California in 1948-49, and at Tulane University in 1950-51, is intended to be both a reference and a text. These objectives are somewhat inconsistent. In particular, as a reference work it offers a reasonably complete coverage of the area, and this has resulted in a more extended treatment than would normally be given in a course. There are many details which are arranged primarily for reference work; for example, I have taken some pains to include all of the most commonly used terminology, and these terms are listed in the index. On the other hand, because it is a text the exposition in the earlier chapters proceeds at a rather pedestrian pace. For the same reason there is a preliminary chapter, not a part of the systematic exposition, which covers those topics requisite to the main body of work that I have found to be new to many students. The more serious results of this chapter are theorems on set theory, of which a systematic exposition is given in the appendix. This appendix is entirely independent of the remainder of the book, but with this exception each part of the book presupposes all earlier developments.Preliminaries
Topological spaces
Moore-smith convergence
Product and quotient spaces
Embedding and metrization
Compact spaces
Uniform spaces
Function spaces
Appendix: Elementary set theory