A First Course in Topology

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As a separate branch of mathematics, topology is relatively young. It was isolated as

a collection of methods and problems by Henri Poincare´ (1854–1912) in his pioneering

paper Analysis situs of 1895. The subsequent development of the subject was dramatic

and topology was deeply influential in shaping the mathematics of the twentieth century

and today.

So what is topology? In the popular understanding, objects like the M¨obius band,

the Klein bottle, and knots and links are the first to be mentioned (or maybe the second

after the misunderstanding about topography is cleared up). Some folks can cite the joke

that topologists are mathematicians who cannot tell their donut from their coffee cups.

When I taught my first undergraduate courses in topology, I found I spent too much time

developing a hierarchy of definitions and too little time on the objects, tools, and intuitions

that are central to the subject. I wanted to teach a course that would follow a path more

directly to the heart of topology. I wanted to tell a story that is coherent, motivating, and

significant enough to form the basis for future study.

To get an idea of what is studied by topology, let’s examine its prehistory, that is,

the vague notions that led Poincar´e to identify its foundations. Gottfried W. Leibniz

(1646–1716), in a letter to Christiaan Huygens (1629–1695) in the 1670’s, described a

concept that has become a goal of the study of topology:

I believe that we need another analysis properly geometric or linear, which treats

PLACE directly the way that algebra treats MAGNITUDE.

Leibniz envisioned a calculus of figures in which one might combine figures with the ease of

numbers, operate on them as one might with polynomials, and produce new and rigorous

geometric results. This science of PLACE was to be called Analysis situs ([Pont]).

We don’t know what Leibniz had in mind. It was Leonhard Euler (1701–1783)

who made the first contributions to the infant subject, which he preferred to call geometria

situs. His solution to the Bridges of K¨onigsberg problem and the celebrated Euler formula,

V −E+F = 2 (Chapter 11) were results that depended on the relative positions of geometric

figures and not on their magnitudes ([Pont], [Lakatos]).