100
Topological Spaccs and Continuous Functions
Ch. 2
Closed Sets and Limit Points
17
101
A , B , and A , denote subsets of a space X . Prove the following:
I f c B , !hen> c B.
K B = A ! B.
( Am > ( J A ,; give an example where equality fails.
J

If t
100
Topological Spaccs and Continuous Functions
Ch. 2
Closed Sets and Limit Points
17
101
A , B , and A , denote subsets of a space X . Prove the following:
I f c B , !hen> c B.
K B = A ! B.
( Am > ( J A ,; give an example where equality fails.
J

If t
Tupulugical Spaces and Continuous Functions
92
Ch. 2
inherits as a subspace of Y is the same as the topology it inherits as a subspace
of X .
2. If 7 and 7' are topologies on X and 7' is strictly finer than 7 , what can you
say about the corresponding sub
170
PmoJ:
Ch. 3
Connectedness and Compactness
Given a collection A of subsets of X, let
be the collection of their complements. Then the following statements hold:
(1) A is a collection of open sets if and only if C is a collection of closed sets.
C of al
280
Complete Metric Spaces and h nction Spaces
Ch. 7
Exercises
1. If X, is metrizable with metric d, then
D (X,Y)= s up(di(xi, y i)/iI
n
i s a metric for the product space X = X,. Show that X is totally bounded
undcr D if each X, is totally bounded under
186
Connectedness and Compactness
Exercises
$29
*Supplcmcntary Exercises: Nets
187
'Supplementary Exercises: Nets
1. Show that the rationals Q are not locally compact.
2. Let cfw_X,] be an indexed family of nonempty spaces.
(a) Show that if X, i s locally
178
Connectedness and Compactness
Ch. 3
(b) Show that R K is connected. [Hint: (m, 0) and (0, CO) inherit their usual
topologles as subspaces of R K.]
(c) Show that WK is not path connected.
4. Show that a connected metric space having more than one poin
82
I opolog~cal paces and Continuous Fum~.lions
S
Ch. 2
where a < b , the topology generated by 8' is called the lower l ima topology on R .
When R is given the lowel limit topology, wc denote it by Wt. Finally let K denote the
set of all numbers of the f
THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MAT3070(20092010)
Sketch of the Solution to homework 7
1. (a) If X = , it is of course compact. Therefore we may assume X = .
Let U be any open cover of X . Since X = , there exists U0 U with
Theorem 1 Let X be a nonempty, compact Hausdorff space with no isolated points.
Then X is uncountable.
Proof. The proof uses the following two Claims.
Claim 2 Let X be Hausdorff without isolated points, and let A X be a closed subset
with nonempty inter