MATH 131: PROBLEM SET 5
Due Friday October 7.
1. Show that connected components of a topological space are closed. Use this to show
that a topological space with nitely many connected components is homeomorphic to
the disjoint union of those nitely many c
MATH 131: PROBLEM SET 3
Due Friday September 23.
1. Let X be a topological space. Let X X be = cfw_(x, x) : x X. ( is called the
diagonal.) Show that X is Hausdorff if and only if is closed in X X.
2. Let J be a set and X be a topological space. Equip the
MATH 131: PROBLEM SET 1
Due Friday September 9.
1. Let X and Y be topological spaces, and let A be a subset of X. Note that A Y can be
endowed with the product topology, where A is given the subspace topology. Note also
that A Y can be endowed with the su
MATH 131: PROBLEM SET 4
Due Friday September 30.
1a. Show multiplication R R R is continuous, i.e. show that the function taking
(a, b) to ab is continuous.
b. Show that if f, g : X R are continuous functions then the function fg, dened
fg(x) = f(x)g(x),
MATH 131: PROBLEM SET 6
Due Friday October 14.
1. Show that a continuous bijection f : X Y with X compact and Y Hausdorff is
a homeomorphism. Give an example to show that such a continuous bijection is not
necessarily a homeomorphism if Y is not assumed t
MATH 131: PROBLEM SET 8
Due Friday November 4.
1. Dene vp : Z cfw_0 Z by vp (pn d) = n where d is not divisible by any factors of p.
|x|p = pvp (x)
for x Z cfw_0 and dene |0|p = 0.
(i) Show that (x, y) |x y|p is a metric on Z.
(ii) Dene Zp to be Zp =
MATH 131: PROBLEM SET 10
Due Friday November 18.
1. Let X be a topological space, and let x0 be a point of X. A continuous function :
[0, 1] X such that (0) = (1) = x0 induces a continuous map S1 = [0, 1]/(cfw_0 cfw_1) X
by the universal property of quoti
MATH 131: PROBLEM SET 11
Due Wednesday November 30.
1. A continuous map f : X Y is said to be proper if for every compact subset K of Y,
f1 K is compact. Show that a proper map f : X Y with X,Y locally compact Hausdorff
is closed, i.e. for any closed set
Math 131: Problem Set 7
1. Denote by Ux a neighborhood of x which can be imbedded in RkUx for some integer kUx .
Then cfw_Ux xX is an open covering of X. Since X is compact, there exists a nite subcover
cfw_U1 , U2 , . . . , Un
MATH 131: PROBLEM SET 7
Due Friday October 28.
1. Let X be a compact Hausdorff space. Suppose that for each x X, there is a neighborhood U of x such that U can be imbedded in RkU for some integer kU . Show that X can
be imbedded in RN for some integer N.
MATH 131: PROBLEM SET 9
Due Friday November 11.
1. Suppose X, Y, and Z are topological spaces, and that Y is Hausdorff and locally
compact. Let C(X, Y) denote the space of continuous functions X Y endowed with the
compact open topology. Show that composit
MATH 131: PROBLEM SET 2
Due Friday September 16.
1. Let X be a topological space and let Y be a subspace of X. Let A Y.
(i) Give a counterexample to the statement
intY (A) = intX (A) Y,
where intY (A) denotes the interior of A taken with respect to the to