proof of Heine-Borel theorem

proof of Heine-Borel theorem

To shorten the proofs, we will be using certain concepts and facts from general
point-set topology, such as the very first assertion below.
Even if you are not familiar with them,
these assertions are all easily proven from the definitions (left as an exercise).

Compact implies closed and bounded.

Since

ℝ

n

is a

Hausdorff space

, any

compact subset

A

has to be closed as well.
That

A

is also

bounded

is very easily seen by taking
the

open cover

{

(

-

k

,

k

)

n

}

k

∈

ℕ

.
Passing to a finite cover, we see that

A

is contained
in a bounded set.
∎

Reduction for closed and bounded implies compact.

To prove that if

A

is closed and bounded implies it is

compact

,
it is only necessary to prove that the rectangle

[

a

,

b

]

n

is compact.
For general

A

, since

A

is bounded, it is contained in some such

compact
rectangle

[

a

,

b

]

n

.
Since

A

is closed and contained in a compact set,

A

is also compact.
∎

The case

n

=

1

: the closed interval is compact..

Let

𝒞

be an arbitrary cover of

[

a

,

b

]

by

open sets

in

ℝ

1

.
Define

S

=

{

x

∈

[

a

,

b

]

:

some finite subcollection of

𝒞

covers

[

a

,

x

]

}

.

Set

t

=

sup

⁡

S

; then

a

≤

t

≤

b

<

∞

.

We first note that the supremum

t

is attained:
that is, there is a finite subcollection of

𝒞

that covers

[

a

,

t

]

.
Obviously we can choose one open set

U

from

𝒞

that covers the point

t

.
This

U

must also cover the

open interval

(

t

-

ϵ

,

t

+

ϵ

)

for some

ϵ

>

0

.
But by the definition of

t

, the interval

[

a

,

t

-

ϵ

]

has a finite subcover.
When this finite subcover is put together with

U

, we obtain a finite subcover for

[

a

,

t

]

.

It is easy to see that

t

cannot be less than

b

.
For the same open set

U

above covers the interval

(

t

,

t

+

ϵ

)

,
so we have a finite subcover for

[

a

,

t

+

ϵ

/

2

]

.
If

t

is the supremum then it has go to be at the right endpoint of the interval

[

a

,

b

]

, i.e.

t

=

b

.
∎

For the case

n

>

1

, apply

Tychonoff

’s Theorem, in the finite case (

http://planetmath.org/ProofOfTychonoffsTheoremInFiniteCase

), which states:

If

X

1

,

…

,

X

n

are each compact,
then

X

1

×

⋯

×

X

n

in the

product topology

is compact.

Here we set

X

i

to be the

closed intervals

; then the general closed rectangle in

ℝ

n

is compact. It is not hard to see that the product topology for the closed rectangle is the same
as its

subspace topology

in the product topology

ℝ

n

=

ℝ

×

⋯

×

ℝ

.
And it is again a standard exercise to show that the product topology on

ℝ

n

is the
same as the norm topology on

ℝ

n

as a

vector space

.

This

completes

the proof of the

Heine-Borel theorem

.

Proof by a bisection argument

There is another proof of the Heine-Borel theorem for

ℝ

n

without
resorting to Tychonoff’s Theorem. It goes by bisecting the rectangle along each of its sides.
At the first stage, we divide up the rectangle

A

into

2

n

subrectangles.
Suppose the open cover

𝒞

of

A

has no finite subcover.
Then one of the subrectangles — call it

A

1

—
must have no finite subcover by

𝒞

. We can subdivide

A

1

into

2

n

pieces; since

A

1

has no finite subcover, one of the new subrectangles of

A

1

also has no finite subcover. And continue
dividing to get a nested sequence of rectangles

A

⊃

A

1

⊃

A

2

⊃

⋯

whose side lengths approach zero, and possessing no finite subcover.

By the

nested interval theorem

, the “limit rectangle”

⋂

i

=

1

∞

A

i

must consist of a sole point

x

,
and this obviously has a finite subcover by an open set

U

∈

𝒞

.
But

U

must contain a small rectangle with centre

x

, which
for

i

large enough, contradicts

A

i

having no finite subcover.

Of course, in both proofs of the Heine-Borel theorem, the completeness of the reals (the least

upper bound

property)
enters in an essential way.

Title

proof of Heine-Borel theorem

Canonical name

ProofOfHeineBorelTheorem

Date of creation

2013-03-22 12:57:40

Last modified on

2013-03-22 12:57:40

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

23

Author

stevecheng (10074)

Entry type

Proof

Classification

msc 54D30

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