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

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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.

https://brainmass.com/math/geometry-and-topology/topology-surfaces-point-set-topology-521373

#### 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.

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## Problems in the Point-Set Topology of R^n

Let Fr(A) denote the frontier set of A and Cl(A) denote the closure of A, where A is a subset of R^n. Solve the following problems.

Exercise 2.6: For any set A, Fr(A) is closed.

Exercise 2.12: If A and B are any sets, prove that Cl(A and B) belongs to Cl(A) and Cl(B). Give an example where Cl(A and B) is empty, but Cl(A) and Cl(B) is not empty.

Exercise 2.17: Show that x is the only limit point of the sequence constructed in Theorem 2.11.

Exercise 2.25: Prove that the composition of two continuous functions is continuous.

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