Homework 2 Solutions
Math 351, Fall 2010
Section 1, Problem 8.
Theorem. Every vector space has a basis.
Let V be a vector space.
Lemma. Let A be an independent subset of V , and suppose that some v V does not belong
to the span of A. Then A cfw_v is indep
Function Spaces
A function space is a topological space whose points are functions. There are many
dierent kinds of function spaces, and there are usually several dierent topologies
that can be placed on a given set of functions. These notes describe thre
Homework 3 Solutions
Math 351, Fall 2010
Section 16, Problem 9.
Proposition. The dictionary order topology on the set R R is the same as the product
topology Rd R, where Rd denotes R in the discrete topology.
Proof. Because singleton sets are a basis for
Homework 1 Solutions
Math 351, Fall 2010
Section 3, Problem 12.
The following table summarizes the properties of the three given orders:
Order Minimum Element Elements Without Immediate Predecessors
(i)
(1, 1)
(1, 1), (2, 1), (3, 1), . . .
(ii)
none
. . .
Homework 5 Solutions
Math 351, Fall 2010
Problem 23.6.
Theorem. Let X be a topological space, and let A, C X. Suppose that C is connected and
C intersects both A and X A. Then C intersects Bd A as well.
Proof. Recall that X is the disjoint union of Int A,
Homework 4 Solutions
Math 351, Fall 2010
Section 19, Problem 8.
Let (a1 , a2 , . . .) and (b1 , b2 , . . .) be sequences of real numbers, where each ai > 0, and dene
a function h : R R by
h(x1 , x2 , . . .) = (a1 x1 + b1 , a1 x2 + b2 , . . .).
Theorem. Th
Homework 6 Solutions
Math 351, Fall 2010
Note: Several of the solutions below use the fact that the metric on a metric space is a
continuous function. Because this is not proven directly in the book, we prove it below.
(You were not required to prove this
Homework 9 Solutions
Math 351, Fall 2010
Problem 55.2.
Theorem. Any nullhomotopic map S 1 S 1 has a xed point.
Proof. Suppose h : S 1 S 1 is nullhomotopic. Then h extends to a map H : B 2 S 1 , which
can be thought of as a map from B 2 to B 2 . By Brouwer
Homework 8 Solutions
Math 351, Fall 2010
Problem 52.2.
Theorem. Let A be a subspace of Rn , let h : (A, a0 ) (Y, y0 ), and suppose that h extends
to a continuous map Rn into Y . Then h is the trivial homomorphism.
Proof. Let H : (Rn , a0 ) (Y, y0 ) be an
Takehome Midterm Solutions
Math 351, Fall 2010
Problem 1
(a)
Theorem. Every discrete subset of R is countable.
We oer two dierent proofs of this theorem.
First Proof. Let D be a discrete subset of R. Let n Z, and consider the intersection
D [n, n + 1]. Th
Takehome Midterm
Math 351, Fall 2010
This midterm has four problems, each of which has several parts. Your solutions must be
A
written up in L TEX, and are due on Sunday, November 14 at midnight. Late solutions
will not be accepted.
Here are the rules for
Homework 7 Solutions
Math 351, Fall 2010
Problem 22.2.
Theorem. Let p : X Y be a continuous map. Suppose there exists a continuous map
f : Y X such that p f is the identity map of Y . Then p is a quotient map.
Proof. Note that p is surjective, since f is