The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
Classication of Surfaces
Richard Koch November 20, 2005
1
Introduction
We are going to prove the following theorem: Theorem 1 Let S be a compact connected 2dimensional manifold, formed from a polygon
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
Math 131
Solution Set 12
Dennis Clark
May 4, 2003
N1
and is isomorphic when restricted to such a neighborhood,
so Y is a covering space. Now, the disk at the center of
Y can be deformation retracted t
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
N1
Math 131
Solution Set 11
Dennis Clark
May 3, 2003
Let X be a M bius band. (a) Show that 1 (X ) = Z.
o
(b) Exhibit a cover p : Y X where Y is simply connected. (c) Show that there does not exist a r
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
Math 131
Solution Set 10
Dustin Cartwright
April 15, 2003
is a bijection. We want to show that this is an isomorphism.
Let f , g be representative paths for two elements of
1 (T , b0 ). Let f and g be
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
(b) A starconvex set is clearly path connected, so it sufces to check triviality of the fundamental group at the star
center. At this point, however, starconvexity ensures that
the straightline hom
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
that for all x, x whenever d (x, x ) < , then d (x, x ) < .
Since (xi ) is Cauchy under d , there is an N such that
for i, j > N , d (xi , x j ) < . But under these same conditions, d (xi , x j ) < .
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
Math 131
Solution Set 7
Dennis Clark
April 15, 2003
4 34 3 Let X be a compact Hausdorff space. Show that X is
metrizable if and only if X has a countable basis.
Assume X is metrizable. X is Lindel f b
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
2
number of rst coordinates in I0 , and so there is some x
[0, 1] such that x y A for all y [0, 1]. This means that
x (0, 1) is an nonempty open subset disjoint from A, so
2
A cannot be dense in I0
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
First, observe that a bijective order preserving map between ordered sets is a homeomorphism of topological
spaces. Also, note that halfopen intervals of R all have
the same order type if theyre open
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
2. Second, if x = a 0, but a = 0, then cfw_(a 1/n
0, a 1/n)  n > 1 forms a countable basis. Since
a
b = 0, c must be less than a. Thus there is some n
such that a 1/n > c. And as in the rst case, e
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
Math 131: Solution Set 3. By Dennis Clark
18 8 Let Y be an ordered set in the order topology. Let
f , g : X Y be continuous.
(a) Show that the set cfw_x f (x) g(x) is closed in X.
(b) Let h : X Y be
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
13 7 Consider the following topologies on R:
Math 131
Solution Set 2
Dustin Cartwright
February 11, 2003
T1 = the standard topology,
T2 = the topology of RK ,
T3 = the nite complement topology,
13 1 L
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
(c) x
Math 131
Solution Set 1
Dustin Cartwright
February 6, 2003
AA
A x A for at least one A A
True, because x A for ever A A . This is equivalent to
the converse above.
The converse is false, becaus
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 431/531, HW 4 SOLUTIONS
Munkres problems, section 22 1. Let p(x) be a if x > 0, b if x = 0 and c if x < 0. Because p1 (a) = (0, ) is open in R, cfw_a is open in A. Because p1 (b) = 0 is closed in
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 531 HOMEWORK 3, SOLUTIONS
18 3. Since i is the identity i1 U = U , so i is continuous if and only if when U is open in T
it is open in T . That is, if and only if T is ner than T . The second pa
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 431/531, SELECTED SOLUTIONS TO HW 2
(1) Show that if A is closed in Y and Y is closed in X then A is closed in X .
Solution: A = Y C where C is closed in X . But Y is also closed in X and the int
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 453
SOLUTIONS TO ASSIGNMENT 11 DECEMBER 3, 2004
Exercise 2 from Section 51, page 330 This is a special case of problem 3 below. Exercise 3 from Section 51, page 330 (a) The formula F(x, t) = (1 
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 453
SOLUTIONS TO ASSIGNMENT 10 NOVEMBER 23, 2004
Exercise 5 from Section 30, page 194 (a) Let D be a countable dense subset of the metrizable space X. I claim that B = cfw_B(x, 1/n)  x D and n Z
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 453
SOLUTIONS TO ASSIGNMENT 9 NOVEMBER 11, 2004
Exercise 6 from Section 28, page 181 X is a metric space, so it is Hausdorff. Thus X is compact Hausdorff and we need only check that f is a contin
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 453
SOLUTIONS TO ASSIGNMENT 8 NOVEMBER 6, 2004
Exercise 2 from Section 26, page 171 (a) Let X be a subspace of R in the finite complement topology, and let cfw_U be an open cover for X. Pick a p
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 453
SOLUTIONS TO ASSIGNMENT 7 NOVEMBER 1, 2004
Exercise 3 from Section 23, page 152 Let B = A A . Then B is connected for each since A and A are connect and have a point in common. Now B = A A an
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 453
SOLUTIONS TO ASSIGNMENT 6 OCTOBER 16, 2004
Exercise 4 from Section 20, page 127 (a) Since each of their component functions are continuous, all three of f , g and h are continuous when R is g
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 453
SOLUTIONS TO ASSIGNMENT 5 OCTOBER 8, 2004
Exercise 4 from Section 18, page 111 We check that f is an imbedding; the argument for g is similar. f is clearly a bijection between X and f (X) = X
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 453
SOLUTIONS TO ASSIGNMENT 4 OCTOBER 7, 2004
Exercise 2 from Section 17, page 100 Using Theorem 17.2, A is closed in Y implies that it is the intersection of Y and a closed subset C of X: A = C
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 453
SOLUTIONS TO ASSIGNMENT 3 SEPTEMBER 26, 2004
Exercise 1 from Sections 1416, page 91 First consider A as a subspace of Y. In this topology, a subset B of A is open if and only if it is of the
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 453
SOLUTIONS TO ASSIGNMENT 2 SEPTEMBER 20, 2004
Exercise 1 from Sections 12 & 13, page 83 We will show that A is open by exhibiting it as a union of open sets. For each x A, let U x be the open
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
MATH 453
SOLUTIONS TO ASSIGNMENT 1 SEPTEMBER 8, 2004
Exercise 4 from Section 3, page 28 (a) Reflexivity, symmetry and transitivity for follow from the corresponding properties for equality, so is an e
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
Midterm EXAM II
Math 5345 Professor: Eliot Brenner November 24, 2008 Name:
by writing my name I swear by the honor code. . .
Read all of the following information before starting the exam: Show all wo
The University of Texas at San Antonio San Antonio
Topology
MATH 312

Fall 2010
Midterm EXAM I
Math 5345 Professor: Eliot Brenner October 6, 2008 Name:
by writing my name I swear by the honor code. . .
Read all of the following information before starting the exam: Show all work,