Math 535 - General Topology
Fall 2012
Homework 2 Solutions
Problem 1. (Brown Exercise 2.4.5) Consider X = [0, 2] \ cfw_1 as a subspace of the real line
R. Show that the subset [0, 1) X is both open and closed in X .
Solution. [0, 1) is open in X because w
Math 535 - General Topology
Fall 2012
Homework 14 Solutions
Problem 1. Let X be a topological space and (Y, d) a metric space. For each compact subset
K X , consider the pseudometric on C (X, Y ) dened by
dK (f, g ) = sup d (f (x), g (x)
xK
and its associ
Math 535 - General Topology
Fall 2012
Homework 13 Solutions
Note: In this problem set, function spaces are endowed with the compact-open topology unless
otherwise noted.
Problem 1. Let X be a compact topological space, and (Y, d) a metric space. Consider
Math 535 - General Topology
Fall 2012
Homework 12 Solutions
Problem 1. Consider the open cover U = cfw_B1 (x)xR of R by open balls of radius 1, i.e.
open intervals B1 (x) = (x 1, x + 1). Find a partition of unity on R subordinate to U .
Solution. We will
Math 535 - General Topology
Fall 2012
Homework 11 Solutions
Problem 1. Let X be a topological space.
a. Show that the following properties of a subset A X are equivalent.
1. The closure of A in X has empty interior: int(A) = .
2. For all non-empty open su
Math 535 - General Topology
Fall 2012
Homework 10 Solutions
Problem 1. Let Top denote the category of topological spaces and continuous maps, and
let CHaus denote the category of compact Hausdor topological spaces and continuous maps.
Show that the Stone-
Math 535 - General Topology
Fall 2012
Homework 9 Solutions
Problem 1.
a. Let X be a topological space with nitely many connected components. Show that each
connected component is open in X .
Solution. Let X = X1 . . . Xn be the partition of X into connect
Math 535 - General Topology
Fall 2012
Homework 8 Solutions
Problem 1. (Willard Exercise 19B.1) Show that the one-point compactication of Rn is
homeomorphic to the n-dimensional sphere S n .
Solution. Note that S n is compact, and a punctured sphere S n \c
Math 535 - General Topology
Fall 2012
Homework 7 Solutions
Problem 1. Let X be a topological space and (Y, d) a metric space. A sequence (fn )nN of
functions fn : X Y converges uniformly to a function f : X Y if for all > 0, there is
an N N satisfying
for
Math 535 - General Topology
Fall 2012
Homework 6 Solutions
Problem 1. Let F be the eld R or C of real or complex numbers. Let n 1 and denote by
F[x1 , x2 , . . . , xn ] the set of all polynomials in n variables with coecients in F.
A subset C Fn of n-dime
Math 535 - General Topology
Fall 2012
Homework 5 Solutions
Problem 1. Let X be a rst-countable topological space, (xn )nN a sequence in X , and y X
a cluster point of this sequence. Show that there is a subsequence (xnk )kN that converges to y .
Solution.
Math 535 - General Topology
Fall 2012
Homework 4 Solutions
Problem 1. Let cfw_ A be a family of directed set. Show that the product
becomes a directed set by dening the relation
A
if in for all A
i.e. the componentwise preorder. (First check that this is
Math 535 - General Topology
Fall 2012
Homework 3 Solutions
Problem 1. Let S 1 R2 be unit circle in the plane, with the subspace topology. Consider
the winding map
f : R S1
t (cos t, sin t).
Show that f induces a homeomorphism R/ S 1 , where the equivalenc