Overview
Problem 1.
Let E and V be disjoint subsets of a metric space (X; d). Assume V is
open in X.
(i) (4 points) Fix x0 2 V . Show that inffd(x0; y); y 2 Eg > 0.
(ii) (3 points) Prove or give a counterexample: inffd(x; y); x 2 V; y 2
Eg > 0.
Problem 2.
Let
X := f(an)n1 R; 9N 2 N such that 8 n N : an = 0g;
and let d1((an)n1; (bn)n1) := supn1 jan bnj:
(a) (3 points) Prove that (X; d1) is a metric space.
(b) (4 points) Is the metric space (X; d1) complete? (Prove your answer).
Problem 3.
i.) (4 points) Let (X; d) be a metric space, and let K1;K2; : : : ;Kn be
a nite family of compact subsets of X. Proved that [ni
=1Ki is a
compact set.
ii.) (3 points) Let (X; d) be a compact metric space. Prove or give a
counterexample: If K1 K2 K3 : : : is an increasing sequence
of compact subsets of X. Then [1n
=1Kn is compact.
Problem 4.
i.) (4 points) Let (X; d) be a metric space. Suppose (xn)n2N is a sequence
in X which converges to x 2 X. Let S := fxn; n 2 Ng [ fxg. Show
that S is compact. (Note: You can use any of the characterizations
of compactness from class).
ii.) (5 points) Let (X; d) be a metric space, let K X be compact and
let C X be closed. Assume that
inffd(x; y); x 2 K; y 2 Cg = 0:
Prove that K \ C 6= ;.