Lee's Introduction to Topological Manifolds is a much better book -- for one, its chapter on CW-complexes will make it much easier to follow Hatcher's Chapter 0. It may also be beneficial to learn other related topics well, including basic abstract algebra, Lie theory, algebraic geometry, and, in particular, differential geometry. This text is designed to provide instructors with a convenient single text resource for bridging between general and algebraic topology courses.

Topology - James R. Munkres - Google Books

in which one would usually read Topology A First Course by Munkres or a similar intro to general topology book, then follow that with something like Algebraic Topology by Hatcher and Differential Topology by Guillemin and Pollack and Milnors Topology from the Differentiable Viewpoint. I'm currently studying Algebraic Topology and Differential Topology (and Differential Geometry) on my own, and I'm thoroughly enjoying it, but currently it seems that Algebraic Topology and Differential Topology, don't use that much General Topology apart from Compactness, Connectedness and the basics.I hope to someday specialize in Algebraic Topology or Differential Topology/Differential Geometry, so would learning more about General Topology have any direct benefit to my studies of these subjects? The latter quarter of the course covers basic notions in algebraic topology (in Munkres, but significantly overlapping with the earliest parts of Hatcher/MAT 560). Images Donate icon An illustration of a heart shape Donate Ellipses icon An illustration of text ellipses. It also deals with subjects like topological spaces and continuous functions, connectedness, compactness, separation axioms, and selected further topics such as function spaces, metrization theorems, embedding theorems and the fundamental group. This intuition is captured by the notion of the fundamental group, which, (very) loosely speaking, is an algebraic object that counts the number of “holes” of a topological space.

Topology: Readings and Homework - Harvard University Topology: Readings and Homework - Harvard University

Princeton has some of the best topologists in the world; Professors David Gabai, Peter Ozsvath and Zoltan Szabo are all well-known mathematicians in their fields. He also includes bibliographic references in the Exercises and Problems for all the original publications of the deeper ones (and much of the other ones too). A popular joke is that for topologists, a doughnut and a coffee mug are the same thing, because one can be continuously transformed into the other. Point-set topology is the subfield of topology that is concerned with constructing topologies on objects and developing useful notions such as separability and countability; it is closely related to set theory.The only point of such a basic, point-set topology textbook is to get you to the point where you can work through an (Algebraic) Topology text at the level of Hatcher.

MIT Mathematics

This introduction to topology provides separate, in-depth coverage of both general topology and algebraic topology. What follows is a wealth of applications—to the topology of the plane (including the Jordan curve theorem), to the classification of compact surfaces, and to the classification of covering spaces. It includes a huge amount of material in the Exercises and Problems as well, which, if presented in full, would make the book unmanageably big. A couple of cheap, but good, books are Point Set Topology by Gaal and Topology for Analysis by Wilansky.After finishing the sequence MAT 365 and MAT 560, topology students can consider taking a junior seminar in knot theory (or some other topic), or, if that is not available, writing a junior paper under the guidance of one of the professors. Optional, independent topics and applications can be studied and developed in depth depending on course needs and preferences. Each of the text's two parts is suitable for a one-semester course, giving instructors a convenient single text resource for bridging between the courses.

Topology student go after Munkres? Where does a Topology student go after Munkres?

In addition, MAT 345 or equivalent comfort with group theory is strongly recommended before enrolling in this course. For the constructivists, some of the proofs might not be up to such framework, the acceptance of the axiom of choice might also irritate others, but all in all, they are well put and quite clear, for the most part. Yes, General Topology is fun and there are many neat old theorems that you will learn by studying it in more detail, but you have to prioratize: Life is short and your time in graduate school is even shorter.

NEW - Greatly expanded, full-semester coverage of algebraic topology—Extensive treatment of the fundamental group and covering spaces. Web icon An illustration of a computer application window Wayback Machine Texts icon An illustration of an open book. However, for a long time, many aspects of 3- and 4-manifolds had evaded study; thus developed the subfield of low-dimensional topology, the study of manifolds of dimension 4 or below. This course is an introduction to algebraic topology, and has been taught by Professor Peter Ozsvath for the last few years.