Official Christmas list thread.

I'm thinking of getting "numbers and proofs" by Allenby, simply because I haven't got the slightest idea how to write proofs and this book looks like a nice introduction

Also @TeeEm any suggestions for books on topology (introductory of course), it looks very interesting indeed. However I've noticed universities don't teach it until 3rd year (at least going by the course content) so finding a book I could understand and appreciate might be like finding a needle in a haystack

x.

@TeeEm has passed on this one, so I'll have a go.

Topology is a very wide field - and the first division you might identify is between General Topology and Algebraic Topology. General topology generalizes the topological notions that we find in good wholesome everyday spaces like $\mathbb{R}^n$ to more general sets (and indeed to some pretty pathological monsters). It tends to get taught in the second year at university after you've understood what the topological aspects of ordinary spaces actually are. There's a very good book by Sutherland called "Introduction to Metric and Topological Spaces" that has been around for donkeys years. But be warned: we do general topology because we have to, not necessarily because we like it! It is not a sexy subject until you get into advanced applications such as fractals.

You've already noted that the fact that universities don't tend to start doing algebraic topology until the third year suggest that it is not really amenable to an elementary treatment. That's very true, it's a subject that builds on what you learn in the first couple of years of university, and represents a pay-off for all that unmotivated slog that you've gone through! So, with that warning in mind, here's a few recommendations:

One of the standard books these days is Allan Hatcher's "Algebraic Topology". It's very good and it's available in a free online version as well as in print. Try reading the first couple of chapters to get a feeling for the subject. Next up is Singer and Thorpe's "Lecture Notes on Elementary Topology and Geometry", which gives a good overview of the subject. Finally Massey's "Algebraic Topology: an Introduction", which is a nice treatment of the beginnings of homotopy theory, but which excludes homology.

Another approach, of course, is to use google and Wikipedia. There's lots of interesting material out there!

"How to prove it" by Velleman is a much better book for proofs and an intro to set theory,

The best intro to topology I know of for minimal prerequisite knowledge is Essential Topology by Crossley. First 6 chapters go through general topology, and the ones after begin algebraic topology, looking at the fundamental group and (much more importantly imo) homology. This stuff will require you to learn about groups first, though.

What are similarities and differences between algebraic and differential geometry?