Unformatted text preview: B 1 , B 2 , , B n , such that
n Bj. X j1 For each j we choose a member U j of the family such that B j U j and, since n Uj X j1 we have found finitely many members of that cover the space X. 3. Suppose that X is a compact metric space and that is a family of closed subsets of X. Prove that if
then there must exist a finite subfamily of such that
We observe first that 182 . X H H X H H X X. Therefore, since X is compact, it is possible to find finitely many members H 1 , H 2 , , H n of the
family such that
n X X Hj j1 and we see at once that
n Hj . j1 4. In a theorem we have just studied we saw that if X is a compact metric space then every sequence in X that
has no more than one partial limit must be convergent.
Is it true that if every sequence in a given metric space having no more than one partial limit must be
convergent, that X must be compact?
Yes, it is true. If every sequence that has no more than one partial limit has to be convergent then
every sequence must have at least one partial limit.
Now let’s change the question. Suppose that we know that, in a given metric space, that every
sequence with precisely one partial limit must be convergent. Must the space be compact. Again
the answer is yes. To see why, suppose that a metric space X fails to be compact and choose a
sequence x n in X such that x n has no partial limit. We now look at the sequence
x 1 , x, x 2 , x, x 3 , x, .
This sequence has precisely one partial limit but is not convergent.
5. Suppose that X is a compact metric space and that S X. Prove that the metric space S is compact if and only
if S is closed in X.
The result follows at once from Theorem 7.12.14
6. Prove that if A and B are compact subspaces of R then A B is also compact. Hint: Show that every sequence in the set A B has a subsequence that converges to a point of A B. Suppose that x n is a sequence in the set A B. For each n, choose numbers a n A and b n B
such that x n a n b n . Using the fact that A is compact, choose a subsequence a n j of a n that
converges to a point a of the set A. For each j we have x n j a n j b n j . Using the fact that B is
compact, choose a subsequence b j i of b n j that converges to a point b of the set B. The
subsequence x n j i of x n converges to the point a b of the set A B. 7. Given a metric space X, is it true that every bounded infinite subset of X has a limit point if and only if every
closed bounded subset of X is compact as a subspace of X?
Yes, this assertion is true.
Suppose that every closed bounded subset of X is compact as a subspace of X and suppose that S
is a bounded infinite subset of X. Since S is compact, every infinite subset of S must have a limit
point. Therefore S must have a limit point.
Suppose that every bounded infinite subset of X has a limit point and that Y is a closed bounded
subset of X. Every infinite subset of Y, being bounded, must have a limit point and, since Y is
closed, any such limit point mus...
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