11.5 Compactness of the interval

11.5 Compactness of the interval

We already pointed out that our constructions of reals are entirely

compatible

with
classical logic. Thus, by assuming the

law of excluded middle

(

LABEL:eq:lem

) and the

axiom
of choice

(

LABEL:eq:ac

) we could develop classical

analysis

, which would essentially
amount to copying any standard book on analysis.

Nevertheless, anyone interested in computation, for example a numerical analyst, ought to
be curious about developing analysis in a computationally meaningful setting. That
analysis in a constructive setting is even possible was demonstrated by

[

Bishop1967

]

.
As a sample of the

differences

and

similarities

between classical and constructive
analysis we shall briefly discuss just one topic—

compactness

of the

closed interval

[

0

,

1

]

and a couple of

theorems

surrounding the

concept

.

Compactness is no exception to the common phenomenon in constructive mathematics that
classically

equivalent

notions bifurcate. The three most frequently used notions of
compactness are:

1.

metrically

compact

:

“Cauchy

complete

and

totally bounded

”,

2.

Bolzano–Weierstraß compact:

“every

sequence

has a

convergent

subsequence

”,

3.

Heine-Borel compact:

“every

open cover

has a finite subcover”.

These are all equivalent in classical mathematics.
Let us see how they fare in

homotopy type theory

. We can use either the Dedekind or the
Cauchy reals, so we shall denote the reals just as

ℝ

. We first recall several basic

definitions

.

Definition 11.5.1

.

A

metric space

(

M

,

d

)

is a set

M

with a map

d

:

M

×

M

→

R

satisfying, for all

x

,

y

,

z

:

M

,

d

⁢

(

x

,

y

)

≥

0

,

d

⁢

(

x

,

y

)

=

d

⁢

(

y

,

x

)

,

d

⁢

(

x

,

y

)

=

0

⇔

x

=

y

,

d

⁢

(

x

,

z

)

≤

d

⁢

(

x

,

y

)

+

d

⁢

(

y

,

z

)

.

Definition 11.5.2

.

A

Cauchy approximation

in

M

is a sequence

x

:

Q

+

→

M

satisfying

∀

(

δ

,

ϵ

)

.

d

⁢

(

x

δ

,

x

ϵ

)

<

δ

+

ϵ

.

The

limit

of a Cauchy approximation

x

:

Q

+

→

M

is a point

ℓ

:

M

satisfying

∀

(

ϵ

,

θ

:

ℚ

+

)

.

d

(

x

ϵ

,

ℓ

)

<

ϵ

+

θ

.

A

complete metric space

is one in which every Cauchy approximation has a limit.

Definition 11.5.3

.

For a

positive

rational

ϵ

, an

ϵ

-net

in a metric space

(

M

,

d

)

is an element of

∑

(

n

:

ℕ

)

∑

(

x

1

,

…

,

x

n

:

M

)

∀

(

y

:

M

)

.

∃

(

k

≤

n

)

.

d

(

x

k

,

y

)

<

ϵ

.

In words, this is a

finite sequence

of points

x

1

,

…

,

x

n

such that every point
in

M

merely is within

ϵ

of some

x

k

.

A metric space

(

M

,

d

)

is

totally bounded

when it has

ϵ

-nets of all
sizes:

∏

(

ϵ

:

ℚ

+

)

∑

(

n

:

ℕ

)

∑

(

x

1

,

…

,

x

n

:

M

)

∀

(

y

:

M

)

.

∃

(

k

≤

n

)

.

d

(

x

k

,

y

)

<

ϵ

.

Remark 11.5.4

.

In the definition of total boundedness we used sloppy notation

∑

(

n

:

N

)

∑

(

x

1

,

…

,

x

n

:

M

)

. Formally, we should have written

∑

(

x

:

List

(

M

)

)

instead,
where

List

⁢

(

M

)

is the inductive type of finite lists from

\autoref

sec:bool-nat.
However, that would make the rest of the statement a bit more cumbersome to express.

Note that in the definition of total boundedness we require pure

existence

of an

ϵ

-net, not mere existence. This way we obtain a

function

which assigns to each

ϵ

:

ℚ

+

a specific

ϵ

-net. Such a function might be called a “

modulus

of
total boundedness”. In general, when porting classical metric notions to homotopy type
theory, we should use propositional

truncation

sparingly, typically so that we avoid
asking for a non-constant map from

ℝ

to

ℚ

or

ℕ

. For

instance

, here is the
“correct” definition of

uniform continuity

.

Definition 11.5.5

.

A map

f

:

M

→

R

on a metric space is

uniformly continuous

when

∏

(

ϵ

:

ℚ

+

)

∑

(

δ

:

ℚ

+

)

∀

(

x

,

y

:

M

)

.

d

(

x

,

y

)

<

δ

⇒

|

f

(

x

)

-

f

(

y

)

|

<

ϵ

.

In particular, a uniformly continuous map has a modulus of uniform continuity,
which is a function that assigns to each

ϵ

a corresponding

δ

.

Let us show that

[

0

,

1

]

is compact in the first sense.

Theorem 11.5.6

.

The closed interval

[

0

,

1

]

is complete and totally bounded.

Proof.

Given

ϵ

:

ℚ

+

, there is

n

:

ℕ

such that

2

/

k

<

ϵ

, so we may take the

ϵ

-net

x

i

=

i

/

k

for

i

=

0

,

…

,

k

-

1

. This is an

ϵ

-net because,
for every

y

:

[

0

,

1

]

there merely exists

i

such that

0

≤

i

<

k

and

(

i

-

1

)

/

k

<

y

<

(

i

+

1

)

/

k

, and so

|

y

-

x

i

|

<

2

/

k

<

ϵ

.

For completeness of

[

0

,

1

]

, consider a Cauchy approximation

x

:

ℚ

+

→

[

0

,

1

]

and let

ℓ

be its limit in

ℝ

. Since

max

and

min

are Lipschitz maps,
the

retraction

r

:

ℝ

→

[

0

,

1

]

defined by

r

(

x

)

:

≡

max

(

0

,

min

(

1

,

x

)

)

commutes
with limits of Cauchy approximations, therefore

r

⁢

(

ℓ

)

=

r

⁢

(

lim

⁡

x

)

=

lim

⁡

(

r

∘

x

)

=

r

⁢

(

lim

⁡

x

)

=

ℓ

,

which means that

0

≤

ℓ

≤

1

, as required.
∎

We thus have at least one good notion of compactness in homotopy type theory.
Unfortunately, it is limited to metric spaces because total boundedness is a metric
notion. We shall consider the other two notions shortly, but first we prove that a
uniformly continuous map on a totally bounded space has a

supremum

,
i.e. an

upper bound

which is less than or equal to all other upper bounds.

Theorem 11.5.7

.

A uniformly continuous map

f

:

M

→

R

on a totally bounded metric space

(

M

,

d

)

has a supremum

m

:

R

. For every

ϵ

:

Q

+

there exists

u

:

M

such
that

|

m

-

f

⁢

(

u

)

|

<

ϵ

.

Proof.

Let

h

:

ℚ

+

→

ℚ

+

be the modulus of uniform continuity of

f

.
We define an approximation

x

:

ℚ

+

→

ℝ

as follows: for any

ϵ

:

ℚ

total
boundedness of

M

gives a

h

⁢

(

ϵ

)

-net

y

0

,

…

,

y

n

. Define

x

ϵ

:

≡

max

(

f

(

y

0

)

,

…

,

f

(

y

n

)

)

.

We claim that

x

is a Cauchy approximation. Consider any

ϵ

,

η

:

ℚ

, so that

x

ϵ

≡

max

⁡

(

f

⁢

(

y

0

)

,

…

,

f

⁢

(

y

n

)

)

and

x

η

≡

max

⁡

(

f

⁢

(

z

0

)

,

…

,

f

⁢

(

z

m

)

)

for some

h

⁢

(

ϵ

)

-net

y

0

,

…

,

y

n

and

h

⁢

(

η

)

-net

z

0

,

…

,

z

m

.
Every

z

i

is merely

h

⁢

(

ϵ

)

-close to some

y

j

, therefore

|

f

⁢

(

z

i

)

-

f

⁢

(

y

j

)

|

<

ϵ

, from which we may conclude that

f

⁢

(

z

i

)

<

ϵ

+

f

⁢

(

y

j

)

≤

ϵ

+

x

ϵ

,

therefore

x

η

<

ϵ

+

x

ϵ

. Symmetrically we obtain

x

η

<

η

+

x

η

, therefore

|

x

η

-

x

ϵ

|

<

η

+

ϵ

.

We claim that

m

:

≡

lim

x

is the supremum of

f

. To prove that

f

⁢

(

x

)

≤

m

for
all

x

:

M

it suffices to show

¬

⁡

(

m

<

f

⁢

(

x

)

)

. So suppose to the contrary that

m

<

f

⁢

(

x

)

. There is

ϵ

:

ℚ

+

such that

m

+

ϵ

<

f

⁢

(

x

)

. But now merely for
some

y

i

participating in the definition of

x

ϵ

we get

|

f

(

x

)

-

f

(

y

i

)

<

ϵ

, therefore

m

<

f

⁢

(

x

)

-

ϵ

<

f

⁢

(

y

i

)

≤

m

, a

contradiction

.

We finish the proof by showing that

m

satisfies

the second part of the theorem, because
it is then automatically a least upper bound. Given any

ϵ

:

ℚ

+

, on one hand

|

m

-

f

⁢

(

x

ϵ

/

2

)

|

<

3

⁢

ϵ

/

4

, and on the other

|

f

⁢

(

x

ϵ

/

2

)

-

f

⁢

(

y

i

)

|

<

ϵ

/

4

merely for some

y

i

participating in the definition of

x

ϵ

/

2

,
therefore by taking

u

:

≡

y

i

we obtain

|

m

-

f

⁢

(

u

)

|

<

ϵ

by

triangle
inequality

.
∎

Now, if in

\autoref

ctb-uniformly-continuous-sup we also knew that

M

were complete, we
could hope to weaken the

assumption

of uniform continuity to

continuity

, and strengthen
the

conclusion

to existence of a point at which the supremum is attained. The usual proofs
of these improvements rely on the facts that in a complete totally bounded space

1.

continuity implies uniform continuity, and

2.

every sequence has a convergent subsequence.

The first statement follows easily from Heine-Borel compactness, and the second is just
Bolzano–Weierstraß compactness.
Unfortunately, these are both somewhat problematic. Let
us first show that Bolzano–Weierstraß compactness implies an instance of excluded middle
known as the

limited principle of omniscience

:
for every

α

:

ℕ

→

𝟐

,

(

∑

n

:

ℕ

α

(

n

)

=

1

𝟐

)

+

(

∏

n

:

ℕ

α

(

n

)

=

0

𝟐

)

.

(11.5.8)

Computationally speaking, we would not expect this principle to hold, because it asks us to decide
whether infinitely many values of a function are

0

𝟐

.

Theorem 11.5.9

.

Bolzano–Weierstraß compactness of

[

0

,

1

]

implies the limited principle of omniscience.

Proof.

Given any

α

:

ℕ

→

𝟐

, define the sequence

x

:

ℕ

→

[

0

,

1

]

by

x

n

:

≡

{

0

if

α

⁢

(

k

)

=

0

𝟐

for all

k

<

n

,

1

if

α

⁢

(

k

)

=

1

𝟐

for some

k

<

n

.

If the Bolzano–Weierstraß

property

holds, there exists a

strictly increasing

f

:

ℕ

→

ℕ

such that

x

∘

f

is a

Cauchy sequence

. For a sufficiently large

n

:

ℕ

the

n

-th term

x

f

⁢

(

n

)

is within

1

/

6

of its limit. Either

x

f

⁢

(

n

)

<

2

/

3

or

x

f

⁢

(

n

)

>

1

/

3

. If

x

f

⁢

(

n

)

<

2

/

3

then

x

n

converges

to

0

and so

∏

(

n

:

ℕ

)

α

⁢

(

n

)

=

0

𝟐

. If

x

f

⁢

(

n

)

>

1

/

3

then

x

f

⁢

(

n

)

=

1

, therefore

∑

(

n

:

ℕ

)

α

⁢

(

n

)

=

1

𝟐

.
∎

While we might not mourn Bolzano–Weierstraß compactness too much, it seems harder to live
without Heine–Borel compactness, as attested by the fact that both classical mathematics
and Brouwer’s Intuitionism accepted it. As we do not want to wade too deeply into general
topology, we shall work with basic

open sets

. In the case of

ℝ

these are the open
intervals with rational

endpoints

. A family of such

intervals

,

indexed by

a type

I

,
would be a map

ℱ

:

I

→

\setof

(

q

,

r

)

:

ℚ

×

ℚ

|

q

<

r

,

with the idea that a pair of rationals

(

q

,

r

)

with

q

<

r

determines the type

\setof

x

:

ℝ

|

q

<

x

<

r

. It is slightly more convenient to allow degenerate intervals as well, so we take a

family of basic intervals

to be a map

ℱ

:

I

→

ℚ

×

ℚ

.

To be quite precise, a family is a dependent pair

(

I

,

ℱ

)

, not just

ℱ

. A

finite family of basic intervals

is one indexed by

\setof

m

:

ℕ

|

m

<

n

for some

n

:

ℕ

. We usually present it by a finite list

[

(

q

0

,

r

0

)

,

…

,

(

q

n

-

1

,

r

n

-

1

)

]

. Finally, a

finite subfamily

of

(

I

,

ℱ

)

is given
by a list of indices

[

i

1

,

…

,

i

n

]

which then determine the finite family

[

ℱ

⁢

(

i

1

)

,

…

,

ℱ

⁢

(

i

n

)

]

.

As long as we are aware of the distinction between a pair

(

q

,

r

)

and the corresponding
interval

\setof

x

:

ℝ

|

q

<

x

<

r

, we may safely use the same notation

(

q

,

r

)

for
both.

Intersections

and

inclusions

of intervals are

expressible

in terms of their
endpoints:

(

q

,

r

)

∩

(

s

,

t

)

:

≡

(

max

(

q

,

s

)

,

min

(

r

,

t

)

)

,

(

q

,

r

)

⊆

(

s

,

t

)

:

≡

(

q

<

r

⇒

s

≤

q

<

r

≤

t

)

.

We say that

(

I

,

λ

⁢

i

.

(

q

i

,

r

i

)

)

(

pointwise

) covers

[

a

,

b

]

when

∀

(

x

:

[

a

,

b

]

)

.

∃

(

i

:

I

)

.

q

i

<

x

<

r

i

.

(11.5.10)

The

Heine-Borel compactness for

[

0

,

1

]

states that every covering family of

[

0

,

1

]

merely has a finite subfamily which still covers

[

0

,

1

]

.

Theorem 11.5.11

.

If excluded middle holds then

[

0

,

1

]

is Heine-Borel compact.

Proof.

Assume for the purpose of reaching a contradiction that a family

(

I

,

λ

⁢

i

.

(

a

i

,

b

i

)

)

covers

[

0

,

1

]

but no finite subfamily does. We construct a sequence of closed
intervals

[

q

n

,

r

n

]

which are nested, their sizes shrink to

0

, and none of them is covered
by a finite subfamily of

(

I

,

λ

⁢

i

.

(

a

i

,

b

i

)

)

.

We set

[

q

0

,

r

0

]

:

≡

[

0

,

1

]

. Assuming

[

q

n

,

r

n

]

has been constructed, let

s

:

≡

(

2

q

n

+

r

n

)

/

3

and

t

:

≡

(

q

n

+

2

r

n

)

/

3

. Both

[

q

n

,

t

]

and

[

s

,

r

n

]

are covered by

(

I

,

λ

⁢

i

.

(

a

i

,

b

i

)

)

, but they cannot both have a finite subcover,
or else so would

[

q

n

,

r

n

]

. Either

[

q

n

,

t

]

has a finite subcover or it does not.
If it does we set

[

q

n

+

1

,

r

n

+

1

]

:

≡

[

s

,

r

n

]

, otherwise we set

[

q

n

+

1

,

r

n

+

1

]

:

≡

[

q

n

,

t

]

.

The sequences

q

0

,

q

1

,

…

and

r

0

,

r

1

,

…

are both Cauchy and they
converge to a point

x

:

[

0

,

1

]

which is

contained

in every

[

q

n

,

r

n

]

.
There merely exists

i

:

I

such that

a

i

<

x

<

b

i

. Because the sizes of the
intervals

[

q

n

,

r

n

]

shrink to zero, there is

n

:

ℕ

such that

a

i

<

q

n

≤

x

≤

r

n

<

b

i

, but this means that

[

q

n

,

r

n

]

is covered by a single interval

(

a

i

,

b

i

)

, while at the same time it has no finite subcover. A contradiction.
∎

Without excluded middle, or a pinch of Brouwerian Intuitionism, we seem to be stuck.
Nevertheless, Heine-Borel compactness of

[

0

,

1

]

can

be recovered in a constructive
setting, in a fashion that is still compatible with classical mathematics! For this to be
done, we need to revisit the notion of cover. The trouble with
(

11.5.10

) is that the truncated existential allows a space to
be covered in any haphazard way, and so computationally speaking, we stand no chance of
merely extracting a finite subcover. By removing the truncation we get

∏

(

x

:

[

0

,

1

]

)

∑

(

i

:

I

)

q

i

<

x

<

r

i

,

(11.5.12)

which might help, were it not too demanding of covers. With this definition we
could not even show that

(

0

,

3

)

and

(

2

,

5

)

cover

[

1

,

4

]

because that would amount
to exhibiting a non-constant map

[

1

,

4

]

→

𝟐

, see

\autoref

ex:reals-non-constant-into-Z. Here we can take a lesson from “pointfree topology”
(i.e. locale theory):
the notion of cover ought to be expressed in terms of open sets, without
reference to points. Such a “holistic” view of space will then allow us to analyze the
notion of cover, and we shall be able to recover Heine-Borel compactness. Locale
theory uses

power sets

,
which we could obtain by assuming propositional resizing;
but instead we can steal ideas from the predicative cousin of locale theory,
which is called “formal topology”.

Suppose that we have a family

(

I

,

ℱ

)

and an interval

(

a

,

b

)

. How might
we express the fact that

(

a

,

b

)

is covered by the family, without referring to points?
Here is one: if

(

a

,

b

)

equals some

ℱ

⁢

(

i

)

then it is covered by the family.
And another one: if

(

a

,

b

)

is covered by some other family

(

J

,

𝒢

)

, and in
turn each

𝒢

⁢

(

j

)

is covered by

(

I

,

ℱ

)

, then

(

a

,

b

)

is covered

(

I

,

ℱ

)

. Notice that we are listing

rules

which can be used to

deduce

that

(

I

,

ℱ

)

covers

(

a

,

b

)

. We should find sufficiently
good rules and turn them into an inductive definition.

Definition 11.5.13

.

The

inductive cover

◁

is a mere

relation

◁

:

(

ℚ

×

ℚ

)

→

(

∑

I

:

𝒰

(

I

→

ℚ

×

ℚ

)

)

→

𝖯𝗋𝗈𝗉

defined inductively by the following rules, where

q

,

r

,

s

,

t

are rational numbers and

(

I

,

F

)

,

(

J

,

G

)

are families of basic intervals:

1.

reflexivity

:

ℱ

(

i

)

◁

(

I

,

ℱ

)

for all

i

:

I

,

2.

transitivity

:

if

(

q

,

r

)

◁

(

J

,

𝒢

)

and

∀

(

j

:

J

)

.

𝒢

(

j

)

◁

(

I

,

ℱ

)

then

(

q

,

r

)

◁

(

I

,

ℱ

)

,

3.

monotonicity:

if

(

q

,

r

)

⊆

(

s

,

t

)

and

(

s

,

t

)

◁

(

I

,

ℱ

)

then

(

q

,

r

)

◁

(

I

,

ℱ

)

,

4.

localization:

if

(

q

,

r

)

◁

(

I

,

ℱ

)

then

(

q

,

r

)

∩

(

s

,

t

)

◁

(

I

,

λ

i

.

(

ℱ

(

i

)

∩

(

s

,

t

)

)

)

.

5.

if

q

<

s

<

t

<

r

then

(

q

,

r

)

◁

[

(

q

,

t

)

,

(

r

,

s

)

]

,

6.

(

q

,

r

)

◁

(

\setof

(

s

,

t

)

:

ℚ

×

ℚ

|

q

<

s

<

t

<

r

,

λ

u

.

u

)

.

The definition should be read as a higher-inductive type in which the listed rules are
point constructors, and the type is

(

-

1

)

-truncated. The first four clauses are of a
general nature and should be intuitively clear. The last two clauses are specific to the

real line

: one says that an interval may be covered by two intervals if they overlap,
while the other one says that an interval may be covered from within. Incidentally, if

r

≤

q

then

(

q

,

r

)

is covered by the empty family by the last clause.

Inductive covers enjoy the Heine-Borel property, the proof of which requires a lemma.

Lemma 11.5.14

.

Suppose

q

<

s

<

t

<

r

and

(

q

,

r

)

◁

(

I

,

F

)

. Then there merely
exists a finite subfamily of

(

I

,

F

)

which inductively covers

(

s

,

t

)

.

Proof.

We prove the statement by

induction

on

(

q

,

r

)

◁

(

I

,

ℱ

)

. There are
six cases:

1.

Reflexivity: if

(

q

,

r

)

=

ℱ

⁢

(

i

)

then by monotonicity

(

s

,

t

)

is covered
by the finite subfamily

[

ℱ

⁢

(

i

)

]

.

2.

Transitivity:
suppose

(

q

,

r

)

◁

(

J

,

𝒢

)

and

∀

(

j

:

J

)

.

𝒢

(

j

)

◁

(

I

,

ℱ

)

. By the

inductive hypothesis

there merely exists

[

𝒢

⁢

(

j

1

)

,

…

,

𝒢

⁢

(

j

n

)

]

which covers

(

s

,

t

)

.
Again by the inductive hypothesis, each of

𝒢

⁢

(

j

k

)

is covered by a finite
subfamily of

(

I

,

ℱ

)

, and we can collect these into a finite
subfamily which covers

(

s

,

t

)

.

3.

Monotonicity:
if

(

q

,

r

)

⊆

(

u

,

v

)

and

(

u

,

v

)

◁

(

I

,

ℱ

)

then we may
apply the inductive hypothesis to

(

u

,

v

)

◁

(

I

,

ℱ

)

because

u

<

s

<

t

<

v

.

4.

Localization:
suppose

(

q

′

,

r

′

)

◁

(

I

,

ℱ

)

and

(

q

,

r

)

=

(

q

′

,

r

′

)

∩

(

a

,

b

)

.
Because

q

′

<

s

<

t

<

r

′

, by the inductive hypothesis there is a finite subcover

[

ℱ

⁢

(

i

1

)

,

…

,

ℱ

⁢

(

i

n

)

]

of

(

s

,

t

)

. We also know that

a

<

s

<

t

<

b

, therefore

(

s

,

t

)

=

(

s

,

t

)

∩

(

a

,

b

)

is covered by

[

ℱ

⁢

(

i

1

)

∩

(

a

,

b

)

,

…

,

ℱ

⁢

(

i

n

)

∩

(

a

,

b

)

]

, which is a
finite subfamily of

(

I

,

λ

⁢

i

.

(

ℱ

⁢

(

i

)

∩

(

a

,

b

)

)

)

.

5.

If

(

q

,

r

)

◁

[

(

q

,

v

)

,

(

u

,

r

)

]

for some

q

<

u

<

v

<

r

then by monotonicity

(

s

,

t

)

◁

[

(

q

,

v

)

,

(

u

,

r

)

]

.

6.

Finally,

(

s

,

t

)

◁

(

\setof

(

u

,

v

)

:

ℚ

×

ℚ

|

q

<

u

<

v

<

r

,

λ

z

.

z

)

by
reflexivity. ∎

Say that

(

I

,

ℱ

)

inductively covers

[

a

,

b

]

when there merely exists

ϵ

:

ℚ

+

such that

(

a

-

ϵ

,

b

+

ϵ

)

◁

(

I

,

ℱ

)

.

Corollary 11.5.15

.

A closed interval is Heine-Borel compact for inductive covers.

Proof.

Suppose

[

a

,

b

]

is inductively covered by

(

I

,

ℱ

)

, so there merely is

ϵ

:

ℚ

+

such that

(

a

-

ϵ

,

b

+

ϵ

)

◁

(

I

,

ℱ

)

.
By

\autoref

reals-formal-topology-locally-compact there is a finite subcover of

(

a

-

ϵ

/

2

,

b

+

ϵ

/

2

)

, which is therefore a finite subcover of

[

a

,

b

]

.
∎

Experience from formal topology shows that the rules for inductive covers are

sufficient

for a constructive

development

of pointfree topology. But we can also provide our own
evidence that they are a reasonable notion.

Theorem 11.5.16

.

1.

An inductive cover is also a pointwise cover.

2.

Assuming excluded middle, a pointwise cover is also an inductive cover.

Proof.

1.

Consider a family of basic intervals

(

I

,

ℱ

)

, where we write

(

q

i

,

r

i

)

:

≡

ℱ

(

i

)

, an interval

(

a

,

b

)

inductively covered by

(

I

,

ℱ

)

, and

x

such that

a

<

x

<

b

.
We prove by induction on

(

a

,

b

)

◁

(

I

,

ℱ

)

that there merely
exists

i

:

I

such that

q

i

<

x

<

r

i

. Most cases are pretty

obvious

, so we show
just two. If

(

a

,

b

)

◁

(

I

,

ℱ

)

by reflexivity, then there merely
is some

i

:

I

such that

(

a

,

b

)

=

(

q

i

,

r

i

)

and so

q

i

<

x

<

r

i

. If

(

a

,

b

)

◁

(

I

,

ℱ

)

by transitivity via

(

J

,

λ

⁢

j

.

(

s

j

,

t

j

)

)

then by
the inductive hypothesis there merely is

j

:

J

such that

s

j

<

x

<

t

j

, and then since

(

s

j

,

t

j

)

◁

(

I

,

ℱ

)

again by the inductive hypothesis there merely
exists

i

:

I

such that

q

i

<

x

<

r

i

. Other cases are just as exciting.

2.

Suppose

(

I

,

λ

⁢

i

.

(

q

i

,

r

i

)

)

pointwise covers

(

a

,

b

)

. By

\autoref

defn:inductive-cover-interval-2 of

\autoref

defn:inductive-cover it
suffices to show that

(

I

,

λ

⁢

i

.

(

q

i

,

r

i

)

)

inductively covers

(

c

,

d

)

whenever

a

<

c

<

d

<

b

, so consider such

c

and

d

. By

\autoref

classical-Heine-Borel
there is a finite subfamily

[

i

1

,

…

,

i

n

]

which already pointwise covers

[

c

,

d

]

, and hence

(

c

,

d

)

. Let

ϵ

:

ℚ

+

be a

Lebesgue number

for

(

q

i

1

,

r

i

1

)

,

…

,

(

q

i

n

,

r

i

n

)

as in

\autoref

ex:finite-cover-lebesgue-number. There is a positive

k

:

ℕ

such that

2

⁢

(

d

-

c

)

/

k

<

min

⁡

(

1

,

ϵ

)

. For

0

≤

i

≤

k

let

c

k

:

≡

(

(

k

-

i

)

c

+

i

d

)

/

k

.

The intervals

(

c

0

,

c

2

)

,

(

c

1

,

c

3

)

, …,

(

c

k

-

2

,

c

k

)

inductively cover

(

c

,

d

)

by repeated use of transitivity and

\autoref

defn:inductive-cover-interval-1
in

\autoref

defn:inductive-cover. Because their widths are below

ϵ

each of
them is contained in some

(

q

i

,

r

i

)

, and we may use transitivity and monotonicity to
conclude that

(

I

,

λ

⁢

i

.

(

q

i

,

r

i

)

)

inductively cover

(

c

,

d

)

. ∎

The upshot of the previous theorem is that, as far as classical mathematics is concerned,
there is no difference between a pointwise and an inductive cover. In particular, since it
is

consistent

to assume excluded middle in homotopy type theory, we cannot exhibit an
inductive cover which fails to be a pointwise cover. Or to put it in a different way, the
difference between pointwise and inductive covers is not what they cover but in the

proofs

that they cover.

We could write another book by going on like this, but let us stop here and hope that we
have provided ample justification for the claim that analysis can be developed in homotopy
type theory. The curious reader should consult

\autoref

ex:mean-value-theorem for
constructive versions of the mean value theorem.

Title

11.5 Compactness of the interval

\metatable

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