## Transcribed Text

Question 3
(a) (i) Let C be a subset of a metric space. Use sequences to state what it means to say
that C is a compact set.
(ii) Let C be a nonempty compact set in the metric space (IR, . I) and let p E R. Show
that the set given by
A={x+p|xEC}, =
is compact.
Note: It is of no importance whether p is an element of C.
(b)
Let (X, d) be a metric space and 0 # C C X.
(i) Let u = (Uili E , 1} be a family of subsets of X. What does it mean to say that u is
an open cover for C? What does it mean to say that U has a finite subcover for C?
(ii) Use open covers to state what it means to say that C is compact.
(iii) Assume that C is a compact set and let p € X \ C. Show, by means of construction,
that there are two disjoint open sets E1 and E2 such that p € E1 and C C E2. You
could support your proof with a graphical illustration.
Hint: Recall that if X # y then there is € > 0 such that Vc(x)nve(y) = 0 and use the
open cover definition of compactness.
(c)
Let (X, d) be a metric space.
(i) What does it mean to say that X is complete?
(ii) We want to show that if X is complete and E C X is closed, then E is complete. For
this, the following proof is suggested:
Let (xn) be a sequence in X. Then since X is complete we have that (xn) is
convergent. Since E is closed we must have that lim Xn = X € E and so E
no+
is complete.
This proof is wrong. Answer briefly the following two questions and then give a correct
proof.
(q1) Shall we start the proof by taking a sequence (xn) in X? If not, then how shall
we start?
(q2) Given that (xn) is a convergent sequence in X does it make sense to say that
since E is closed then lim Xn = X € E? Explain.
n->oo

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