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
it
disjoint open sets, for instance U = (, 2) and V = ( 2, ). The connected subsets
are just points, for if a connected subset C contain
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)
becomes
d(b, b) + d(b, c) d(b, c)
Now d(b, b) = 0 by axiom one, so this gives
Mathematics 431/531 Midterm
October 31, 2005
Name
1. Dene:
(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
open.
(b) product topology on X Y , where X and Y are topological spaces
An
Mathematics 431/531 Final Examination
December 7, 2005
Name
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 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