John Lee 
Introduction to Topological Manifolds [PDF ebook] 

This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition.Although this second edition has the same basic structure as the first edition, it has been extensively revised and clarified; not a single page has been left untouched. The major changes include a new introduction to CW complexes (replacing most of the material on simplicial complexes in Chapter 5); expanded treatments of manifolds with boundary, local compactness, group actions, and proper maps; and a new section on paracompactness.This text is designed to be used for an introductory graduate course on the geometry and topology of manifolds. It should be accessible to any student who has completed a solid undergraduate degree in mathematics. The author’s book Introduction to Smooth Manifolds is meant to act as a sequel to this book.