# 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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#### Solution Summary

Four problems in topology are solved that involve stereographic projection of the sphere into the plane, compactness, connectedness, and the fixed-point set of a mapping.