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Define a totally bounded metric space.

For every e>0, the space can be written as a finite union of
sets with diameter less than e.

page 140 in palace

page 140 in palace

What is the name for a metric space satisfying:

For every e>0, the space can be written as a finite union of sets with diameter less than e.

For every e>0, the space can be written as a finite union of sets with diameter less than e.

a totally bounded metric space

Prove that a compact metric space is totally bounded.

Suppose that the space is not totally bounded.

Construct a sequence which has no convergent subsequence.

Construct a sequence which has no convergent subsequence.

Prove that if every sequence contained in a metric space has a
Cauchy subsequence, then this space is totally bounded.

Suppose that the space is not totally bounded.

Construct a sequence which has no Cauchy subsequence.

page 141 in palace

Construct a sequence which has no Cauchy subsequence.

page 141 in palace

Prove that if a metric space is totally bounded, then every
sequence contained in this space has a Cauchy subsequence.

Notice that for a given e>0, every infinite set has an infinite
subset with diameter less than e. Use this fact to construct a
Cauchy subsequence of any sequence.

page 141 in palace

page 141 in palace

What is a finitely bound collection of sets?

Let W be a collection of sets.

W is finitely bound iff its every finite subcollection has nonempty intersection.

W is finitely bound iff every finite intersection of sets in W is nonempty.

W is finitely bound iff its every finite subcollection has nonempty intersection.

W is finitely bound iff every finite intersection of sets in W is nonempty.

What does it mean that a collection of sets has the finite
intersection property?

Let W be a collection of sets.

(Every finite subcollection of W has non-empty intersection.)

W has the finite intersection property iff every finite intersection of sets belonging to W has nonempty intersection.

(1) W is a collection of sets with the finite interec

(Every finite subcollection of W has non-empty intersection.)

W has the finite intersection property iff every finite intersection of sets belonging to W has nonempty intersection.

(1) W is a collection of sets with the finite interec

Prove that in a sequentially compact metric space every
finitely bound collection of closed sets has nonempty
intersection.

page 142 in golden gate

Let W be a finitely bound collection of closed subsets of a sequentially compact metric space X. We know that the space is totally bounded. Hence, for every natural n, X is a finite union of balls of radius 1/n. Suppose that for each of these balls there is a set in W which has empty intersection with this ball. By the nature of W, this finite collection of

Let W be a finitely bound collection of closed subsets of a sequentially compact metric space X. We know that the space is totally bounded. Hence, for every natural n, X is a finite union of balls of radius 1/n. Suppose that for each of these balls there is a set in W which has empty intersection with this ball. By the nature of W, this finite collection of

If there is a point belonging to the intersection of a
collection of sets, we say that this collection has ___________.

it has nonempty intersection

When no point belongs to the intersection of a collection of
sets, we say that the collection has ___________.

it has empty intersection