Homework 8, due Tuesday, November 27, 2012
Do any 4 of the 6 problems. Each problem is worth 25 points. Solutions will be graded for
correctness, clarity and style.
(1) Let X be a topological space and let Y be a set. Let f : X Y be a map,
not necessarily
Homework 7, due Thursday, November 1, 2012
Do any 4 of the 5 problems. Each problem is worth 25 points. Solutions will be graded for
correctness, clarity and style.
(1) Let R2 be the graph of a function f : R R. If R and R2 are given the standard
topologi
Homework 6, due Thursday, October 25, 2012
Do any 5 of the 8 problems. Each problem is worth 20 points. Solutions will be graded for
correctness, clarity and style.
(1) Let (X, dX ) and (Y, dY ) be metric spaces. Dene d : X Y R by
d(x1 , y1 ), (x2 , y2 )
Homework 5, due Thursday, October 11, 2012
Do any 5 of the 8 problems. Each problem is worth 20 points. Solutions will be graded for
correctness, clarity and style.
(1) Let X be a topological space. If C is a nite subset of X , show that C is compact.
Sol
Homework 4, due Thursday, September 20, 2012
Given sets X and Y , dene the product X Y to be the set X Y = cfw_(x, y ) : x X, y Y ,
and dene the projection maps X : X Y X and Y : X Y Y by X (x, y ) = x
and Y (x, y ) = y for all (x, y ) X Y .
Now assume TX
Homework 3, due Thursday, September 13, 2012
Let X be a metric space with metric d. Let S X . We say that S is bounded if for some
r > 0 and x X we have S DX (x, r).
If X is a topological space and S X , note that Cl(S ) = S if S is closed. I.e., the clos
Homework 2, due Thursday, September 6, 2012
Let f : X Y be a mapping of sets. We say f is one to one (or injective) if whenever x1
and x2 are in X with f (x1 ) = f (x2 ) then x1 = x2 . Another way to say it is: f is injective if
x1 = x2 always implies f (
Homework 1 Solutions
Let X and Y be sets and let f : X Y be a mapping. Let A, B X and C, D Y be subsets, and let
Ai X , i I , be a family of subsets, indexed by some set I .
Remember: the goal in writing proofs is not only to be right, but to be understoo
Exam 2, Thursday, November 6, 2012
Do any 8 of the 13 problems. Each problem is worth 12 points. (For each True/False problem,
answer T if it is true, and if it is false, answer F and give an explicit counterexample.)
(1) True or False: Let A and B be sub
Exam 1, Thursday, September 27, 2012
Do any 10 of the 14 problems. Each problem is worth 10 points. (For each True/False problem, answer T if it is true, and if it is false, answer F and give an explicit counterexample.)
(1) True or False: Every subset of