Math 8161 Homework 4
Due Thursday, 10/15/09
1. Let cfw_Ai be a family of connected subsets of a topological space X, such that Ai Aj =
for all i, j. Show that i Ai is connected.
2. Let A X be a connected set, and suppose that A Y A. Prove that Y is conn
Math 8161 Homework 2
Due Tuesday, 9/22/09
1. Let A and B be subsets of a topological space X. Prove or disprove the following:
a) A B = A B.
b) A B = A B.
2. Let X be a topological space, and let be the diagonal of the product X X. That is,
= cfw_(x, x)
Math 8161 Homework 1
Due Thursday, 9/10/09
1. Suppose (X, D) is a metric space. Which of the following functions are metrics on X?
a) D1 (x, y) = k D(x, y), where k is a positive real number.
b) D2 (x, y) = k D(x, y), where k is any real number.
c) D3 (x,
Math 8161 Homework 6
Due Thursday, 11/12/09
1. Let X be a metric space, with the property that every point x X has a metric neighborhood
B (x) with compact closure. Let G be a group that acts freely and properly on X. Prove that
the quotient map p : X X/G
Math 8161 Homework 5
Due Thursday, 10/29/09
1. Let M be a manifold. Prove that the connected components of M are exactly the same as
its path components, and that each of those components is itself a manifold.
2. Let SL(n, R) be the set of n n matrices wi
Math 8161 Homework 7
Due Tuesday, 12/8/09
1. Let X = S 2 A, where A is an axis connecting the north and south poles of S 2 . Describe
the universal cover X of X and the action of 1 (X) on the universal cover.
2. Problem 2 on the last homework implies that
Math 8161 Homework 7
Due Tuesday, 11/24/09
1. Prove that there are no retractions r : X A for each of the following cases:
a) X = R3 and A is any knot (embedded circle) in R3 .
b) X = S 1 D 2 and A is its boundary torus S 1 S 1 .
c) X = S 1 D 2 and A is t
Math 8161 Homework 3
Due Thursday, 10/1/09
1. Find a metric space X and a subset A X that is closed and bounded, but not compact.
2. Let X be a Hausdor space, and let A, B be disjoint compact subsets of X. Prove that
there exist disjoint open sets U and V