MAT 441C
HW #7
4/9/13 (due Friday 4/12/13 at 5:30 pm)
25 points
Name
Justify all answers. Unsupported answers may not receive full credit.
1. Let M be the 1-sphere S 1 = cfw_(x, y) R2 | x2 + y 2 = 1. Let T M be the subspace of M R2 R4
consisting of pairs
MAT 441C HW #4 Name w
2/27/13 (due Wednesday 3/6)
25 Juaty I m. 0mm when my an: nun- (III audit.
1. (a)SupposeXisatopologicalspaoe.nndAisaconnectedsubspaceofX. SuppoeeAngz.
ProveBisconnected.
Suppose 6 (5 A01 (onM'FeA, nen Exm\'$<I\)4O-f are open
Sub
w
as. . t a b
MAT 441C I-IW #5 Name E {mote
3/7/13 (due Monday 3/11 at 5:30 p111)
25 pomts Justify all answars. Unsupported answers Inny not, receive fun credit
E. Let X r {1,1} >< [«1,1]. (So X is a union of two disjoint ne segments.) Let N the the
e
MAT 441C
4/1/13 (due Friday 4/5/13)
25 points
HW #6
Name
Justify all answers. Unsupported answers may not receive full credit.
1. Let X and Y be topological spaces, and suppose Y is compact.
(a) Let x0 X, and let W be an open subset of X Y containing cfw_
MAT 441C HW #6 Name
4/1/13 {due Friday 4/5/13)
25 pomi-S Justify n11 z-mswnrg. Unsunperted answers may nor receive fuil credit
1. Let X and Y be topological spaces, and suppese Y is compact.
Let :39 E X , and iet EV be an open subset of X X Y containi
MAT 441C
2/27/13 (due Wednesday 3/6)
25 points
1.
HW #4
Name
Justify all answers. Unsupported answers may not receive full credit.
(a) Suppose X is a topological space, and A is a connected subspace of X. Suppose A B A.
Prove B is connected.
1
(b) Prove t
MAT 441C
HW #5
3/7/13 (due Monday 3/11 at 5:30 pm)
25 points
Name
Justify all answers. Unsupported answers may not receive full credit.
1. Let X = cfw_1, 1 [1, 1]. (So X is a union of two disjoint line segments.) Let the the
equivalence relation on X dene
MAT 441C
2/9/13 (due Wednesday 2/13)
25 points
HW #3
Name
Justify all answers. Unsupported answers may not receive full credit.
1. Let f : X Y be a continuous function. Let (f ) X Y be the graph of f , dened by
(f ) = cfw_(x, y) X Y | y = f (x), considere
MAT 441C
3/15/2013
140 points
Exam 2
Name
1.(30) (a) Suppose X is an innite set, endowed with the discrete topology. Prove that X is not
compact.
(b) Let Y be the set of integers, with the conite topology.1 Prove that Y is compact.
(c) Give an example of
MAT 441C
4/19/2013, due Wednesday 4/24
160 points
Exam 3 (Take-home)
Name
Take-home exam rules: you may use texts and notes from lecture, but are not to discuss the exam with any human, by any means (corporeal or
electronic), except the instructor. No the
MAT 441C
1/28/13 (due Monday 2/4)
25 points
HW #2
Name
Justify all answers. Unsupported answers may not receive full credit.
1. A topological space is is called a T1 space if cfw_x is a closed set for each x X.
(a) Prove: X is a T1 space if and only if, f
MAT 441C
1/18/13 (due Friday 1/25/13)
25 points
HW #1
Name
Justify all answers. Unsupported answers may not receive full credit.
1. Let X Rn . Show that the open -ball BX (x, ) is an open subset of X, for any x X and any
> 0.
2. A topological space is a s