## MAT530 Final Examination - 3 hours - December 13, 1995

Each part of each question is worth 10 points. Total = 110.
1 Give an example of a continuous, injective map f: R --> R^2 which is not a homeomorphism onto its image.

2 Prove that in a normal space, given two disjoint closed sets A,B, there exist disjoint, closed sets A', B' with A contained in the interior of A' and B contained in the interior of B'.

3 a Prove that if A,B are disjoint, compact sets in R^2, then (*) there exist points a in A and b in B such that d(a,b) < = d(a',b') for any other such pair of points. (d is the Euclidean distance).

b Same problem, but now suppose A,B are closed, and only B is known to be compact. Prove (*).

c Show by an example that (*) fails to hold under the weaker hypothesis that A,B are closed.

4 Consider the set of all functions f: R --> [0,1]. Since a function is determined by its values at every point, this set may be identified with [0,1] times [0,1] times ..., one factor for each t in R. Give this set the product topology, and call the resulting space F.

a Give an example (not the whole space!) of a neighborhood of the function sin^2 t in F.

b Consider the set B contained in F of functions which are everywhere < = 1/2. Show that B is closed in the product topology.

5 a Suppose X is an arc-connected space, and that p: Y --> X is a covering map. Suppose one point x in X has k pre-images. Prove that every point of X has k pre-images. (Such a map is called a k-fold cover).

b Exhibit two non-homeomorphic, connected 3-fold covers of the Figure-8 space. Prove carefully that each is a covering and that they are not homeomorphic.

6 In R^3 with coordinates x,y,z, let S be the unit circle about the origin in the (x,y)-plane.

a Prove that R^3 - {z-axis} has S as deformation retract.

b Prove or disprove: R^3 - S has the z-axis as deformation retract.