550A HW 3, 2/20/14
§18 #1, 8, 11
§18
Exercise 1 open
Let f 2 1R; } R be a continuous function with the 6 ~ 5 denition. Let U C R Then for any x) 6 U ,
by the denition of open and basis (open intervals), 3(ambz) C R such that f (as) E (ambz) C U. Let
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HOMEWORK 6
Problem 5.1. Prove that Rn is not homeomorphic to R if n > 1 (Hint:
consider what happens if you delete a point from each space).
Proof. Suppose by way of contradiction that Rn is homeomorphic to R.
Then by denition there exists a homeomorphism
Math 550A, Homework 5
Midterm 1, Metric spaces and Connectedness
Due in class, Tuesday, 3/18
Reading Read 20 and 23 of Munkres.
Exercises (to be done on your own)
1. Prove that the collection of all -balls in a metric space actually forms a basis for
a to
Math 550A, Homework 4
Arbitrary Products
Due in class, Thursday, 3/6
Reading Read 19 of Munkres.
Problems (to turn in)
1. Prove that if cfw_X J is a collection of Hausdor spaces, then
dor in both the product and box topologies.
2. Munkres 19, exercise 7.
Math 550A, Homework 2
Closed sets, T1 and Hausdor spaces
Due in class, Thursday, 2/13
Reading Read 17 of Munkres.
Exercises (to do on your own)
1. Munkres 17, exercise 16 (only for R, R and RK ).
2. Munkres 17, exercise 3
3. Prove: a subspace of a Hausdor
Math 550A, Homework 3
Continuous Functions
Due in class, Thursday, 2/20
Reading Read 18 of Munkres.
Exercises (to do on your own)
1. Show that the subspace (a, b) R with a < b is homeomorphic to (0, 1).
2. Dene S 1 to be the following subset of R2 :
S 1 =
Math 550A, Homework 4 Solutions
April 2, 2014
Problem 1. Prove that if cfw_X J is a collection of Hausdor spaces, then
is Hausdor in both the product and box topologies.
J
X
Proof. Let cfw_X J be a collection of Hausdor spaces. Consider (x )J = (y )J
in J
MATH 550A, HOMEWORK 1
BASIS; ORDER, PRODUCT, AND SUBSPACE TOPOLOGIES
Due at start of class, Tuesday, 2/4
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 t