Final Exam
Math 500, Fall 2011
INSTRUCTIONS:
This exam is due in my mailbox or my inbox by noon on December 20th.
You may use your textbook (Munkres), your own homework assignments, your class notes,
the rst midterm, and the second midterm but NO OTHER
MATH 500, HOMEWORK 1
BASIS; ORDER, PRODUCT, AND SUBSPACE TOPOLOGIES
Due at start of class, Thursday, 9/22
Reading. Read 1316 of Munkres.
Exercises (to do on your own).
(1) If X is a set, show that the collection of all one-point sets cfw_x is a basis for
Math 500, Homework 2
Closed sets, T1 and Hausdor spaces, continuous functions
Due at start of class, Tuesday, 10/4
Reading Read 17 18 of Munkres.
Exercises (to do on your own)
1. Munkres 17, exercise 16 (only for R, R and RK ).
2. Prove: a product of two
Math 500, Homework 3
General Product Topology
Due at start of class, Thursday, 10/6
Reading Read 19 of Munkres.
Exercises (to do on your own)
1. Prove that if cfw_X J is a collection of Hausdor spaces, then
in both the product and box topologies.
Problems
Math 500, Homework 4
Metric spaces and Connectedness
Due at start of class, Tuesday, 11/1
Reading Read 20 21 and 2628 of Munkres.
Exercises (to do on your own)
1. Prove that the collection of all -balls in a metric space actually forms a basis for a
topol
Math 500, Homework 5
Compactness
Due in class, Tuesday, 11/7
Reading 2628
Exercises (to do on your own)
1. Does every topological space have a nite cover?
2. Prove that the unit n-sphere S n is compact.
3. (First, review lim inf of a sequence of real numb
Math 500, Homework 6
Paths, homotopies, and the fundamental group
Due Thursday, 11/30
Reading 51, 52
Exercises (to do on your own)
1. Prove that a group G has a unique identity element. Prove that a group element
g G has a unique inverse.
2. Let G and H b
Math 500, Homework 7
Covering spaces, fundamental group
Due at start of class, Thursday, 12/8
Reading 53-55
Exercises (to do on your own)
1. Find a covering p : E B of the gure-eight B such that p1 (b) consists of three
points for each b.
2. Let X be the