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Topological Surfaces

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1) The definitions of surface (in terms of gluing panels) and what it means for two surfaces to be topologically equivalent.

2) A description of the three features of surfaces that characterize them in terms of their topology.

3) Three examples of pairs of surfaces that agree on two of the features but differ on the third one. For example, an annulus and a torus are both ______ have ______ equal to 0, but differ in ______.

4) An explanation of whether a T-shirt and pair of sweat pants (no zipper) are topologically equivalent.

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A surface is a two dimensional object. This time our bug can crawl along the surface in two independent directions at most points. In principle, a surface can be built up by gluing little square panels together. The squares might have to be very little and you may need lots of them to build a surface that has no visible corners, but from a topological point of view corners can be smoothed out. Two surfaces S1 and S2 are equivalent if there is a "nice" bijection between ...

Solution Summary

Topological surfaces: definitions, examples of features of surfaces. Proving that a pair of sweat pants and a T-shirt are not topological equivalent.

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Topology of Surfaces: Point-Set Topology in R^n.

I have these problems from Topology of Surfaces by L.Christine Kinsey: the problems I require assistance with are 2.26, 2.28, 2.29, and 2.32. These are stated below.

PROBLEM (Exercise 2.26). Describe what stereographic projection does to

(1) the equator,
(2) a longitudinal line through the north and south poles,
(3) a triangle drawn on the punctured sphere.

PROBLEM (Exercise 2.28). Show that compactness is a topological property (as defined in [1, Definition 2.22]), and give examples to show that closedness and boundedness are not.

PROBLEM (Exercise 2.29). Prove that X is connected if and only if X cannot be written as a union of two on-empty disjoint sets which are open relative to X.

PROBLEM (Exercise 2.32). Let f : R^n → R^n be a continuous function. Show that the fixed point set for f , F(f ) = {t ∈ R^n | f (t) = t}, is closed.

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