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Pure Math

Topology

What is Topology?

Topology deals with “the properties that are preserved through deformations, twistings, and stretchings of objects”. Without delving into too many specifics: it deals with manifolds, which you can stretch and distort in any way, as long as you do not make any rips or glue any parts of the manifold together. Specifically, surfaces are a type of manifold. They are made of 2-dimensional shapes but exist in 3-d. You can think of this like creating a paper cube – it’s made out of something 2-dimensional (and is thus “hollow”) but it is still 3-dimensional.

Surfaces

Think about a circle. Topologically, since we are allowed to stretch and distort it, it is the same as an oval (think about grabbing it with your hands and pulling them apart) or even a square (pinching 4 points and pulling apart to make a square). A circle with a hole in it is even the same as a cylinder (pull the hole up to the top and pinch the outer edge of the circle down)

We can get lots of interesting results when we think about surfaces topologically. One other curious structure is the Mobius strip, which you can get if you glue the ends of a paper strip together with one twist. This results in a surface that has only one side! Try drawing a line on the front side of a Mobius strip until it reaches itself – this should never reach the back side. But you’ll find that the back side is the front side – this is one of the curious properties of the mobius strip.

Point-Set and Algebraic Topology

Topology comes in two different flavors, point-set and algebraic topology. Typically, one starts out with point-set topology and moves on to algebraic

Prerequisites

Topology is a challenging undergraduate course. Point-set topology does not really have prereqs other than “mathematical maturity” and being comfortable with proofs. Basic set theory is also helpful but not necessarily required.

Resources

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