Mathematics 431/531 Final Examination
December 7, 2005
1. (a) Give the denition of a connected topological space.
Answer: A topological space X is connected if it is impossible to write X = U V
where U and V are nonempty, disjoint, open subsets of X
Assignment 9; Due Wednesday, November 30
12.10a Every map : [0, 1] X is continuous. So we can always nd a path from p to q
by letting (t) = p for t < 1 and (1) = q.
12:10f The set looks like the picture below. The set A is the series of innitely many rays
Assignment 8; Due Friday, November 18
10.7b The rst part of this proof is exactly the rst part of the proof in the book. The
line Lx divides A into two pieces of equal area and divides B into pieces of area b1 (x) and
b2 (x), where b1 is the area of the p
Assignment 7; Due Friday, November 11
9.8 a The set Q is not connected because we can write as a union of two nonempty
disjoint open sets, for instance U = (, 2) and V = ( 2, ). The connected subsets
are just points, for if a connected subset C contain
Assignment 6; Due Friday, November 4
8.2a If X is nite, then every set is open and X has the discrete topology. Thus if x = y ,
the open sets U = cfw_x and V = cfw_y separate x and y .
Conversely, suppose X is Hausdor and let x = y . Choose disjoint U an
Assignment 5; Due Friday, October 28
6.6a Imagine that X Y has some unspecied topology.
Suppose X Y X is continuous and let U X be open. The inverse image of this set
is U Y ; by continuity this set is open.
Similarly if X Y Y is continuous and V Y is ope
Assignment 4; Due Friday, October 21
5.3b Note that A Y is closed if and only if Ac Y is open. By denition of the
quotient topology, this happens if and only if f 1 (Ac ) is open, which happens if and only
if the complement of this set, which equals f 1 (
Assignment 3; Due Friday, October 14
2.8a The closure of cfw_1, 2, 3, . . . is itself because the set is already closed. Indeed the complement is a union of open intervals. The closure of the set of rationals is all of R because every real number is a lim
Assignment 2; Due Friday, October 7
1.5a For each xed x, must prove that whenever > 0, there is > 0 such that if
d(p, x) < , then |f (p) f (x)| < . Since f (x) = d(x, y ), this last inequality can be written
|d(p, y ) d(x, y )| < .
The idea of the rest of
Assignment 1; Due Friday, September 30
1.2: The triangle inequality must hold for every choice of a, b, and c. For instance, it must
hold if a = b, so
d(a, b) + d(a, c) d(b, c)
d(b, b) + d(b, c) d(b, c)
Now d(b, b) = 0 by axiom one, so this gives
December 2, 2005
Hausdor Spaces, Continued
Theorem 1 A compact subset of a Hausdor space is closed.
Theorem 2 Let : X Y be onto; give Y the quotient topology. If X is compact
Hausdor and is a closed map, then Y is Hausdor.
October 27, 2005
It is important to know all of the denitions listed in this review sheet.
Ive also listed the important theorems. Any listed theorem might appear on the midterm,
but many of these theorems have very ea
Mathematics 431/531 Midterm
October 31, 2005
(a) continuous map between topological spaces
Answer: A map f : X Y is continuous if whenever U Y is open, f 1 (U ) X is
(b) product topology on X Y , where X and Y are topological spaces
MATH 431/531 FALL 2016
DUE SEPTEMBER 30, 2016.
INSTRUCTOR: ROBERT LIPSHITZ
Reminder: homework is due at the beginning of class, handed to me or in
my mailbox on the second floor of Fenton.
Note. This homework assignment is intended partly to ma