Notes on Topology

These are links to (mostly) PostScript files containing notes for various topics in topology. They were originally written back in the 1980's, then revised around 1999. Eventually I'd like to revise them, at which point they'll be available as web pages and PDF files like my other online notes.

I learned topology at M.I.T. from Topology: A First Course by James Munkres. At the time, the first edition was just coming out; I still have the photocopies we were given before the printed version was ready!

Hence, I'm a bit biased: I still think Munkres' book is the best book to learn from. The writing is clear and lively, the choice of topics is still pretty good, and the exercises are wonderful. Munkres also has a gift for naming things in useful ways (The Pasting Lemma, the Sequence Lemma, the Tube Lemma).

(I like Paul Halmos's suggestion that things be named in descriptive ways. On the other hand, some names in topology are terrible --- "first countable" and "second countable" come to mind. They are almost as bad as "regions of type I" and "regions of type II" which you still sometimes encounter in books on multivariable calculus.)

I've used Munkres both of the times I've taught topology, the most recent occasion being 1999. The lecture notes below follow the order of the topics in the book, with a few minor variations.

Here are some areas in which I decided to do things differently from Munkres:

• I prefer to motivate continuity by recalling the epsilon-delta definition which students see in analysis (or calculus); therefore, I took the pointwise definition as a starting point, and derived the inverse image version later.
• I stated the results on quotient maps so as to emphasize the universal property.
• I prefer Zorn's lemma to the Maximal Principle; for example, in constructing connected components, I use Zorn's lemma to construct maximal connected sets.
• I did Tychonoff's theorem at the same time as the other stuff on compactness. The proof for a finite product is long enough that I decided to save some time by omitting it and just doing the general case.
• It seems simpler to me to do Urysohn's lemma by indexing the level sets with dyadic rationals in [0,1] rather than rationals.

• funct.ps (86,912 bytes; 5 pages): Review of topics in set theory.
• top.ps (168,720 bytes; 6 pages): Topological spaces, bases.
• closed.ps (55,509 bytes; 4 pages): Closed sets and limit points.
• haus.ps (132,025 bytes; 2 pages): Hausdorff spaces.
• contin.ps (100,021 bytes; 3 pages): Continuous functions.
• homeo.ps (83,891 bytes; 3 pages): Homeomorphisms.
• prod.ps (51,962 bytes; 3 pages): The product topology.
• Metric spaces
[PDF]
• quot.ps (116,395 bytes; 5 pages): Quotient spaces.
• conn.ps (82,400 bytes; 4 pages): Connectedness.
• realconn.ps (55,934 bytes; 3 pages): Connected subsets of the reals.
• pathco.ps (45,136 bytes; 3 pages): Path connectedness, local connectedness.
• compact.ps (240,878 bytes; 8 pages): Compactness (including Tychonoff's theorem).
• compreal.ps (69,020 bytes; 3 pages): Compact subsets of the reals.
• limcomp.ps (107,691 bytes; 4 pages): Limit point compactness and sequential compactness.
• loccpt.ps (70,399 bytes; 3 pages): Local compactness; the one-point compactification.
• countab.ps (40,734 bytes; 2 pages): The countability axioms.
• sep.ps (139,569 bytes; 5 pages): The separation axioms: regularity and normality.
• urysohn.ps (69,043 bytes; 2 pages): Urysohn's lemma.
• tietze.ps (45,448 bytes; 3 pages): The Tietze Extension Theorem.