some properties of uncountable subsets of the real numbers

some properties of uncountable subsets of the real numbers

Let

S

be an

uncountable

subset of

ℝ

.
Let

𝒜

:=

{

(

x

,

y

)

:

(

x

,

y

)

∩

S

⁢

is countable

}

. For

ℝ

is hereditarily Lindelöff, there is a

countable

subfamily

𝒜

′

of

𝒜

such that

⋃

𝒜

′

=

⋃

𝒜

. For the reason that each of members of

𝒜

′

has a countable

intersection

with

S

, we have that

(

⋃

𝒜

′

)

∩

S

is countable. As the

open set

⋃

𝒜

′

can be expressed uniquely as the union of its components, and the components are countably many, we label the components as

{

(

a

n

,

b

n

)

:

n

∈

ℕ

}

.

See that

(

⋃

𝒜

′

)

∩

S

is precisely the set of the

elements of

S

that are NOT the

condensation points

of

S

.

Now we’d propose to show that

{

a

n

,

b

n

:

n

∈

ℕ

}

is precisely the set of the points which are

unilateral

condensation points of

S

.

Let

x

be a unilateral (left, say) condensation point of

S

. So, there is some

r

>

0

with

(

x

,

x

+

r

)

∩

S

countable. So, there is some

(

a

n

,

b

n

)

such that

(

x

,

x

+

r

)

⊆

(

a

n

,

b

n

)

. See, if

x

∈

(

a

n

,

b

n

)

, then

x

is NOT a condensation point, for

x

has a

neighbourhood

(

a

n

,

b

n

)

which has a countable intersection with

S

. But

x

is a condensation point; so,

x

=

a

n

. Similarly, if

x

is a right condensation point, then

x

=

b

n

.

Conversely

, each

a

n

⁢

(

b

n

,

resp

)

is a left (right, resp) condensation point. Because, for each

ϵ

∈

(

0

,

b

n

-

a

n

)

, we have

(

a

n

,

a

n

+

ϵ

)

∩

S

countable. And as no

a

n

,

b

n

is in

⋃

𝒜

′

,

a

n

,

b

n

are condensation points.

So,

⋃

𝒜

′

is the set of non-condensation points - it is countable; and

{

a

n

,

b

n

}

are precisely the unilateral condensation points. So, all the rest are bilateral condensation points. Now we see, all but a countable number of points of

S

are the bilateral condensation points of

S

.

Call

T

the set of all the bilateral condensation points that are IN

S

. Now, take two

x

<

y

in

T

. As

x

is a bilateral condensation point of

S

,

(

x

,

y

)

∩

S

is uncountable; and as

T

misses atmost countably many points of

S

,

(

x

,

y

)

∩

T

is uncountable. So,

T

is a subset of

S

with in-between property.

We summarize the moral of the story: If

S

is an uncountable subset of

ℝ

, then

1.

The points of

S

which are NOT condensation points of

S

, are at most countable.

2.

The set of points in

S

which are unilateral condensation points of

S

, is, again, countable.

3.

The bilateral condensation points of

S

, that are in

S

, are uncountable; even, all but countably many points of

S

are bilateral condensation points of

S

.

4.

The set

T

⊆

S

of all the bilateral condensation points of

S

has got the property: if

∃

x

<

y

∈

T

, then there is also

z

∈

T

with

x

<

z

<

y

.

Title

some properties of uncountable subsets of the

real numbers

Canonical name

SomePropertiesOfUncountableSubsetsOfTheRealNumbers

Date of creation

2013-03-22 16:40:42

Last modified on

2013-03-22 16:40:42

Owner

sauravbhaumik (15615)

Last modified by

sauravbhaumik (15615)

Numerical id

21

Author

sauravbhaumik (15615)

Entry type

Topic

Classification

msc 54F65

Classification

msc 54F05

Classification

msc 12J15

Classification

msc 54E35

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