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MATH 403
EXAM 2
November 11, 2014
This exam contains 5 problems.
You will have 75 minutes to take the exam.
No books1 notes, or calculators are allowed on the exam.
Problem 2 3 4 5 TOT
Name & Section:
Homework 1, MATH 403, Fall 2015
Due Friday August 28 by 4 PM.
Only starred problems are graded.
(1)* Prove that
3 is irrational.
(2)* Browder 1.10.6
(3)* Browder 1.10.18
(4)* In additi
MATH 403 ANALYSIS I - SPRING 2010
SOLUTIONS to HOMEWORK 5
Problem 1. Let f : (X, d) (Y, ). Show that the following are equaivalent:
(a) f is continuous.
(b) f (A) f (A) for every A X .
(c) f 1 (B ) (f
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lVIath 403 Homework 5
due Tuesday. 10/ 14
)r any set X the dismte meme on X is define as p(:i:,y) = 1 if :1: 3% U and
Mr, 31) :- ii if r z y. Verify that p is a distance function.
3 tt (X. p) be any
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MATH 403H
PROBLEM SET #17
Vector spaces
Definition. A nonempty set L with two operations, addition + and
multiplication by scalars R, is called vector space if it satises the following
three groups of
PROBLEM SET #4
Minimality of the Cantor sets
Cantor sets can be obtained not only as subsets of [0, 1], but in many
other ways as well.
Definition. Let (X, d) and (Y, ) be two metric spaces. Then
the
PROBLEM SET #6
Points of discontinuity and the Baire Category Theorem
Let us consider the following two functions:
Q (x) =
1
0
if x is rational,
if x is irrational,
called the Dirichlet function, and
PROBLEM SET #7
Contraction mapping principle
Definition. Let (X, d) be a metric space. A map f : X X is said to
be contracting if there exists < 1 such that for any x, y X
d(f (x), f (y) d(x, y).
(1)
MATH 403H
PROBLEM SET #8
Definition. A subset C of a metric space (X, d) is said to be compact
if any open cover of K has a nite subcover, that is whenever cfw_Oi | i I
is a collection of open sets of
MATH 403H
PROBLEM SET #10
Connectedness
Definition. A subset E of a metric space X is said to be disconnected
if there are disjoint open subsets U1 and U2 in X such that
E U1 U2 , E U1 = , E U2 = .
A
MATH 403H
PROBLEM SET #9
Continuity and compactness
66. Let X be a compact metric space and f a continuous map of X onto
a metric space Y . Prove that Y = f (X) is itself compact.
Solution. Let cfw_V
MATH 403H
PROBLEM SET #11
Connected and path-connected sets
82. The continuous image of a path-connected space X is path-connected.
Solution. Let f : X Y be continuous and surjective; take any two poi
MATH 403H
PROBLEM SET #12
Universality of the Hilbert cube
We proved in 29 that any perfect complete metric space X contains a
closed subset homeomorphic to the Cantor set. This means that the Cantor
MATH 403H
PROBLEM SET #13
108. The Sierpinski Carpet S is obtained from the unit square by removing the middle-ninth square (1/3, 2/3) (1/3, 2/3), then removing
from each square (i/3, (i + 1)/3) (j/3,
MATH 403H
PROBLEM SET #14
The spaces of continuous functions
Let (X, d) and (Y, ) be two metric spaces, fn : X Y be a sequence of
functions. When does this sequence converge to a limit? We can conside
MATH 403H
PROBLEM SET #16
Stone-Weierstrass Approximation Theorem
Our goal is to prove that any f C 0 ([a, b], R) can be uniformly approximated by polynomials.
126. Prove that for any continuous funct
MATH 403H
PROBLEM SET #15
120. Let X be a compact metric space X, and A be a dense in X. Prove
that for any > 0 there is a nite subset cfw_a1 , . . . , ak A which is -dense
in X in the sense that eac
PROBLEM SET #5
33. Prove that a map f : X Y is continuous if and only if for each open
set U Y its inverse image f 1 (U ) is open in X.
Solution. Let x0 f 1 (U ). Then f (x0 ) U , and since U is open,