Overview
Problem 1. Give examples of a metric space (X; d) and:
(a) an open set that is not closed,
(b) a closed set that is not open,
(c) a set (di erent from ;;X) that is both open and closed.
(d) a set that is not open, nor closed.
(In particular: Not open 6) closed!!!)
Problem 2. Let E be a subset of a metric space (X; d). Show that int(int(E)) = int(E).
Problem 3. Exercise 1.2.4. from the book.
Problem 4. Let (X; d) be a metric space, and A;B X.
(a) Show that A [ B = A [ B.
(b) Similarly as in (a), show that for any family of sets (Ai)i2I , we have
[i2IAi [i2IAi:
Give an example in (R; j:j) to show that the other inclusion doesn't hold in general.
(c) Show that A \ B A \ B.
(d) Find examples of nonempty A;B in (R; j:j) such that A \ B = ;, yet A = B. (In particular, the
inclusion in (c) can be strict!)
Problem 5. Let (X; d) be a metric space, and E X. Show that
E =
\
fB j E B and B is closedg:
Remark: You can show this in essentially two di erent ways: using sequences, or the original de nitions
with open sets. It is a good exercise for yourself to try both!
Problem 6. Give an example of a metric space (X; d) and a (necessarily in nite) family of open sets
An X such that \n2NAn is not open and not closed.
Problem 7. Let X be a set. Suppose d1 and d2 are two di erent metrics on X and there exists constants
c; k > 0 such that
8x; y 2 X : c d1(x; y) d2(x; y) k d1(x; y):
Let (xn)1n
=1 be a sequence in X. Show that limn!1 xn = x in (X; d1) if and only if limn!1 xn = x in
(X; d2).
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