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hw2
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One-page homework sheet from a topology course, MATH 4181 section 001, Fall 1999, kept in the Royster topology notes folder. Problems include proving that a scaled metric is a metric, the p-adic-style metric on the integers, continuity of the integral on continuous functions under the integral metric, and computing distances and the diameter of the unit disk in the usual, taxicab, maximum and discrete metrics.
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MATH 4181 001 Fall 1999
Problem Set 2
1. (Exercise 1, page 34) Let ( X;d) be a metric space and let k >0 be any positive real
number. Let dk(x;y) =kd(x;y) for any two points x;y2X. Show that ( X;dk) is a
metric space.
2. (Exercise 8, page 35) Let Zbe the set of integers. Let pbea positive prime integer.
Given distinct integers m;n2Zthere is a unique integer t=t(m;n) such that
m n=ptk, wherekis an integer not divisible by p. De ne a function d:ZZ!R
by
d(m;n) =(
0 ifm=n
1
ptifm6=n
Prove that (Z;d) is a metric space. Let p= 3. What is the set of elements n2Zsuch
thatd(0;n)<1? What is the set of elemnets m2Zsuch thatd(0;m)<1
3?
3. (Exercise 1, page 39) Let Xbe the set of continuous functions f: [a;b]!R. Letd
be the metric on Xgiven by
d(f;g) =Zb
ajf(t) g(t)jdt;
forf;g2X. For each element f2Xset
I(f) =Zb
af(t)dt:
Prove that this function I: (X;d)!(R;d) is continuous.
4. ForP= ( 2;1) andQ= (3;4) inR2, compute the distance from PtoQin each of
the following metrics:
(a) usual;
(b) taxicab;
(c) maximum;
(d) discrete.
5. LetB=fP= (x1;x2)2R2jx2
1+x2
21g. Compute the diameter of Bin each of the
following metrics:
(a) usual;
(b) taxicab;
(c) maximum;
(d) discrete.
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