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Show that a totally bounded metric space can be written as a
finite union of open balls with radius e, for every e>0.

______

Define an interesting metric on P(|N).

The one I picked up from the book "Topological Spaces" during vacation 2000.

In this item, do not prove that it is indeed a metric.

The one I picked up from the book "Topological Spaces" during vacation 2000.

In this item, do not prove that it is indeed a metric.

A,B c |N

if (A = B) then d(A,B) = 0

if (A != B) then d(A,B) = 1/min(A+B)

A+B = (A\B)u(B\A)

page 151 in golden gate

if (A = B) then d(A,B) = 0

if (A != B) then d(A,B) = 1/min(A+B)

A+B = (A\B)u(B\A)

page 151 in golden gate

A,B c N

if (A = B) then d(A,B) = 0

if (A != B) then d(A,B) = 1/min(A+B); A+B = (A\B)u(B\A)

Show that (P(N),d) is a metric space.

if (A = B) then d(A,B) = 0

if (A != B) then d(A,B) = 1/min(A+B); A+B = (A\B)u(B\A)

Show that (P(N),d) is a metric space.

First show that

(*) d(A,B) < 1/m <=> A n [1,m] = B n [1,m].

Using this show the triangle inequality.

page 151 in palace

(*) d(A,B) < 1/m <=> A n [1,m] = B n [1,m].

Using this show the triangle inequality.

page 151 in palace

Let (X,d) and (Y,g) be metric spaces. Let f : X -> Y.

What does it mean that f is an isometry of X into Y?

What does it mean that f is an isometry of X into Y?

/\(a,b:-X) [ d(a,b) = g(f(a),f(b)) ]

(it follows that an isometry is 1-1)

(I som' it rEE)

(it follows that an isometry is 1-1)

(I som' it rEE)

Let (X,d) and (Y,g) be metric spaces.

What does it mean that they are isometric?

What does it mean that they are isometric?

There exists a function f : X -> Y such that

(1) d(a,b) = g(f(a),f(b)) for all a,b from X

(2) f(X) = Y

(this f is bijective)

isometric (I suh met' rik)

(1) d(a,b) = g(f(a),f(b)) for all a,b from X

(2) f(X) = Y

(this f is bijective)

isometric (I suh met' rik)

Let (X,d) be a metric space. Consider the set B(X,|R) of
bounded functions f:X->|R.

Let ||f|| = sup {|f(x)|:x:-X}. Show that B(X,|R) is a metric space with d(f,g) = ||f-g||.

Let ||f|| = sup {|f(x)|:x:-X}. Show that B(X,|R) is a metric space with d(f,g) = ||f-g||.

_______

Let (X,d) be a metric space. Let B(X,|R) be the set of bounded
functions f:X->|R. Equip it with the sup metric. What kind of
metric space is this B(X,|R) ?

it is a complete metric space

page 154 in the palace notebook

page 154 in the palace notebook

Consider the set of all real-valued sequences a[n] such that
/\n 0<=a[n]<=1/n.

Equip this set with the sup metric.

What kind of metric space is this?

Equip this set with the sup metric.

What kind of metric space is this?

It is compact. PROOF: Call it K and show that it is a closed
subset of the set of all bounded real-valued sequences.
Conclude that it is complete. Next show that K is totally
bounded. Here's how. Take any n:-|N.

Let G = {g:-K : /\k:-|N g(k):-{0,1/n,2/n,...,(n-1)/n,1}}.

Show that G is finite.

Show that the collection of balls with center g:-G and ra

Let G = {g:-K : /\k:-|N g(k):-{0,1/n,2/n,...,(n-1)/n,1}}.

Show that G is finite.

Show that the collection of balls with center g:-G and ra

Let X = (0,oo). Define d(x,y) = |x-y| / (x*y). (X,d) is a
metric space.

Is d similar to the Euclidean metric?

Is d similar to the Euclidean metric?

Yes.

Use d(x,y) = |1/x - 1/y|.

Use d(x,y) = |1/x - 1/y|.

Show that in a complete metric space (X,d), A c X:

A is precompact <=> clo(A) is compact.

Demonstrate that completeness is essential here.

A is precompact <=> clo(A) is compact.

Demonstrate that completeness is essential here.

In every metric space

A is precompact <=> clo(A) is precompact

In a complete metric space, a closed set is complete.

Precompact and complete implies compact.

A is precompact <=> clo(A) is precompact

In a complete metric space, a closed set is complete.

Precompact and complete implies compact.