K. Jänich, Topology, Undergraduate Texts in Mathematics, Springer, Berlin (1984), [ISBN:9780387908922]. Introduction

 I have not been able to find a free copy of the book online.

Topology is one subject that I did not study as an undergraduate studying Mathematics with Philosophy at Oxford. I've regretted it ever since. It seems every branch of higher mathematics depends on an undergraduate knowledge of algebra, analysis and topology, and I've been playing catch-up since then. It's time to sit down and work through a textbook steadily.

The Introduction begins with Jänich's claim that science is not divided into tiny specialities, each with just a handful of experts, which is sometimes the stereotype. Rather, modern science is very interconnected - every branch of science uses results from every other branch of science (he seems to just assume that mathematics is a part of science). I agree, but I would add this was true before 'modern' science - it always amazes me how many different branches of knowledge people like Maxwell contributed to. Jänich was writing in 1984, but the point still holds today.

He is motivating the study of topology as an abstract subject that generalizes many structures from different branches of mathematics, and allows us to apply the notion of "closeness" and our geometric intuition in these different areas. I had not seen this point made before; if anything, usually the impression given is that we must be suspicious of our geometric intuition when working in topology, because there are weird topological spaces out there where our expectations from nice Euclidean space lead us astray.

He then gives a brief overview of the historical development of the field, tracing it back to Cantor's work and making the very valid point that the definition of a topological space could not have been made before the 20th century's set theory-based view of mathematics. It still seems strange to think that humans have been doing mathematics for thousands of years, but most of the subjects that mathematicians now think of as mathematics would have been incomprehensible to anybody before at least 1850.

Next: The Introduction over with, next time we begin the actual mathematical work with Chapter I - Fundamental Concepts.


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