Mini-Series: Topology
The following posts provide an introduction to topology that requires only some basic (”naive”) set theory as a background.
- Introduction
- Metric Spaces and Euclidean n-Space
- Open Sets in Metric Spaces
- Mappings of Metric Spaces
- Topological Spaces
- Homeomorphisms
- The Subspace Topology
- Product Spaces
- Quotient Spaces
- Surfaces By Identification
- Connected Sums and Polygonal Presentations
- Triangulability and Orientatability
- Connectedness and Path-Connectedness
You can get most of the definitions you need at a bargain price from
Mendelson, Bert, Introduction to Topology
although it’s a little dry as only the final chapter is geometrically-flavoured. If you don’t mind a bit of trigonometry and similar bits of traditional geometry, a very nice introduction is
Penrose, Jordan and Huggett, A Topological Aperitif
Some of the later material of this mini-series is covered in much greater rigour in
Lee, John M, Introduction to Topological Manifolds
which is a really beautiful book, but requires a fairly good grasp of group theory and a willingness to work as it’s sometimes a bit terse.
A perennial favourite “second course” on topology is this one, which is available free online:
Hatcher, Alan, Algebraic Topology
It requires at least as much from you as Lee does, and assumes you know a bit more than this mini-series contains, but Hatcher has a great way of choosing examples that are both colourful and instructive. We’ll very probably do a mini-series on some of this material later.
A bit of an outlier this, but for old-fashioned, very nerdy point-set topology we like
Hocking and Young, Topology
Notice it doesn’t have the word “introduction” in the title. Read Mendelson first, but again this book is full of fascinating examples and bizarre spaces. For even more of that sort of thing, see the classic
Steen and Seebach, Counterexamples in Topology
Both this and Hocking and Young are seductively cheap, but make sure you’re prepared for them; today topology tends to be much more geometrically-motivated, like the mini-series we wrote, whereas this stuff comes from an age when every point counted.
[Update: Yes, we were planning to do a final instalment on compactness and in the end we didn’t. We’ll probably come back to that in a one-off post when we’ve found a nice way to do the “finite subcovers” thing in a nice way.]