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Let X = [0,1). Define d(x,y) = |x/(1-x) - y/(1-y)|.

Show that (X,d) is a metric space.

Show that (X,d) is a metric space.

f : [0,1) -> [0,oo)

f(x) = x / (1-x)

Show that f is 1-1.

Conclude that d is a metric.

f(x) = x / (1-x)

Show that f is 1-1.

Conclude that d is a metric.

Let X = [0,1). Define d(x,y) = |x/(1-x) - y/(1-y)|. (X,d) is a
metric space.

Is (X,d) complete?

Is (X,d) complete?

Yes.

f : [0,1) -> [0,oo)

f(x) = x / (1-x)

Use the fact that f is a bijection.

f : [0,1) -> [0,oo)

f(x) = x / (1-x)

Use the fact that f is a bijection.

Consider an infinitely countable complete metric space.

How many isolated points does it have?

How many isolated points does it have?

Infinitely many.

If you remove an isolated point from a complete metric space,

then the remaining space is still complete.

Use the Baire Category Theorem to obtain one isolated point.

If you remove an isolated point from a complete metric space,

then the remaining space is still complete.

Use the Baire Category Theorem to obtain one isolated point.

Consider a complete metric space which has no isolated points.

Can it be countable?

Can it be countable?

NO.

A countable complete metric space must have an isolated point.

(see the previous item)

A countable complete metric space must have an isolated point.

(see the previous item)

Let A be a closed subset of |R. Suppose that it is infinite and
has no isolated points. Can A be countable?

NO.

A closed subset of |R is complete.

A countable complete set must have an isolated point.

(see the previous item)

A closed subset of |R is complete.

A countable complete set must have an isolated point.

(see the previous item)

Let (X,d) be a metric space. A c X.

What does it mean that A is nowhere-dense?

What does it mean that A is nowhere-dense?

Int(Clo(A)) = O

(the interior of the closure of A is empty)

The closure of A contains no ball.

nowhere-dense = rare

(the interior of the closure of A is empty)

The closure of A contains no ball.

nowhere-dense = rare

What do you call a subset of a metric space whose closure
contains no ball?

1) nowhere-dense

2) rare

2) rare

Let (X,d) be a metric space. A c X.

What does it mean that A is meager?

What does it mean that A is meager?

A meager set is a countable union of rare sets.

(The closure of a rare set contains no ball.)

(Every rare set is meager.)

(The closure of a rare set contains no ball.)

(Every rare set is meager.)

Let (X,d) be a metric space. A c X.

What does it mean that A is rare?

What does it mean that A is rare?

Int(Clo(A)) = O

(the interior of the closure of A is empty)

The closure of A contains no ball.

rare = nowhere-dense

(the interior of the closure of A is empty)

The closure of A contains no ball.

rare = nowhere-dense

What do you call a subset of a metric space that is a countable
union of rare sets? (A rare set is one whose closure contains
no ball.)

1) meager

2) of the first category

2) of the first category