Math 527 Metric and Topological Space
Homework 4
Name: Zhaoning Yang
September 20, 2013
Problem 1
in X.
(Munkres P.100 #2) Show that if A is closed in Y and Y is closed in X, then A is closed
Solution:
Proof. Let Y be a subspace of X. Since A is closed in
Math 527 Metric and Topological Space
Homework 5
Name: Zhaoning Yang
September 27, 2013
Problem 1 (Munkres P.118 #3) Prove Theorem 19.4: If each space X is a Hausdor space , then
X is a Haudor space in both the box and product topologies.
Solution:
Proof.
Math 527 Metric and Topological Space
Homework 8
Name: Zhaoning Yang
October 18, 2013
Problem 1 (Munkres P.163 #10) Let X be a space. Let us dene x y if there is no separation
X = A B of X into disjoint open sets such that x A and y B.
(1) Show this relat
Math 527 Metric and Topological Space
Homework 2
Name: Zhaoning Yang
September 6, 2013
Problem 1
(Munkres, P.51 #1) Show that Q is countably innite.
Solution:
Proof. Let f : Z (Z cfw_0) Q dened by f (m, n) = m/n. So f is clearly surjective. By the result
Math 527 Metric and Topological Space
Homework 3
Name: Zhaoning Yang
September 13, 2013
Problem 1 (Munkres P.83 #1) Let X be a topological space; let A be a subset of X. Suppose that for
each x A there is an open set U such that U A. Show that A is open i
Math 527 Metric and Topological Space
Homework 6
Name: Zhaoning Yang
October 4, 2013
Problem 1
(Munkres P.133 #3) Let Xn be a metric space with metric dn for n Z+ .
(1) Show that (x, y) = max1in cfw_di (xi , yi ) is a metric for the product space
n
i=1 Xi
Math 527 Metric and Topological Space
Homework 9
Name: Zhaoning Yang
October 25, 2013
Problem 1
(Munkres P.177 #2) Let X be a metric space with metric the d; let A X be nonempty
(1) Show that d(x, A) = 0 x A.
(2) Show that if A is compact, d(x, A) = d(x,
Math 527 Metric and Topological Spaces
Homework 11
Name: Zhaoning Yang
November 8, 2013
32 Normal Spaces
Exercise 4
Solution:
Show that every regular Lindelf space is normal.
o
We rst a proof a preliminary result before the main step.
Lemma 1. Let X be a
Math 527 Metric and Topological Space
Homework 7
Name: Zhaoning Yang
October 11, 2013
Problem 1 (Munkres P.152 #1) Let T and T be two topologies on X. If T T, what does connectedness of X in one topology imply about the connectedness in the other?
Solutio
Math 527 Metric and Topological Space
Homework 10
Name: Zhaoning Yang
November 1, 2013
Problem 1 (Munkres P.181 #6) Let (X, d) be a metric space. If f : X X satises the condition
d(f (x), f (y) = d(x, y) for all x, y X, then f is called an isometry of X.
Math 527 Metric and Topological Space
Homework 1
Name: Zhaoning Yang
September 3, 2013
Problem 1 (Munkres, P.28, #1) Dene two points (x0 , y0 ) and (x1 , y1 ) of the plane to be equivalent if
y0 x2 = y1 x2 . Check that this is an equivalent relation and d
Math 527 Metric and Topological Space
Homework 12
Name: Zhaoning Yang
Due: Friday, November 15, 2013
Problem 1.
(Munkres 35.#1.) Show that the Tietze extension theorem implies the Urysohn lemma.
Solution:
Proof. Suppose the Tietze extension theorem. Let X