AbstractWe show that if a computable sequence of real-valued functions on an effectively compact metric space converges pointwise monotonically to a computable function, then the sequence converges effectively uniformly to the function. This is an effectivized version of Dini's Theorem

Kamo, Hiroyasu

Elsevier - Publisher Connector 

Effective Dini's Theorem on Effectively Compact Metric Spaces 

Eﬀective Dini’s Theorem on Eﬀectively
Compact Metric Spaces
Hiroyasu Kamo1
Dept. of Information and Computer Sciences, Faculty of Science,
Nara Women’s University,
Nara, Japan
Abstract
We show that if a computable sequence of real-valued functions on an eﬀectively compact metric
space converges pointwise monotonically to a computable function, then the sequence converges
eﬀectively uniformly to the function. This is an eﬀectivized version of Dini’s Theorem.
Keywords: Dini’s Theorem, eﬀectively compact metric space, computable function, eﬀective
uniform convergence
1 Introduction
If a sequence of real-valued continuous functions on a compact space con-
verges pointwise monotonically to a continuous function, then the sequence
converges uniformly to the function. It is called Dini’s theorem and one of the
fundamental theorems in functional analysis and general topology.
From the viewpoint of computability, the question arises: whether we can
eﬀectivize Dini’s theorem, in other words, whether there is a theorem which is
a Dini’s theorem with some topological concepts replaced by their computa-
tional counterparts. In this paper, we show a positive answer to this question
in the case of metric spaces, more precisely, the theorem that if a computable
sequence of real-valued functions on an eﬀectively compact metric space con-
1 Email: wd@ics.nara-wu.ac.jp
Electronic Notes in Theoretical Computer Science 120 (2005) 73–82
1571-0661      © 2005 Elsevier B.V. 
www.elsevier.com/locate/entcs
doi:10.1016/j.entcs.2004.06.035
Open access under CC BY-NC-ND license. 
verges pointwise monotonically to a computable function, then the sequence
converges eﬀectively uniformly to the function.
Meanwhile, if a computable sequence of real numbers converges monoton-
ically to a computable real number, then the sequence converges eﬀectively to
the real number. It is called the monotonic convergence theorem [6]. The main
theorem in this paper is not only an eﬀectivization of Dini’s thorem but also
an extension of the monotonic convergence theorem to real-valued continuous
functions on eﬀectively compact metric spaces. The monotonic convergence
theorem is considered a special case of our theorem on C({0}), the space of
all real-valued continuous functions on a singlton.
2 Preliminaries
For a relation S ⊂ X1 × · · · ×Xm × Y1 × · · · × Yn and a tuple (x1, . . . , xm) ∈
X1×· · ·×Xm, the set {(y1, . . . , yn) ∈ Y1×· · ·×Yn | (x1, . . . , xm, y1, . . . , yn) ∈ S}
is denoted by S(x1, . . . , xm).
We ﬁx some standard tuple functions and corresponding projection func-
tions on N. 〈-, . . . , -〉 denotes the n-tuple function. (-)nk denotes the cor-
responding kth projection function. We often identify an n-tuple sequence
(xk1,...,kn) with its serialization (x(k)n1 ,...,(k)nn)k∈N.
We use the terminology and the notation on computability of real numbers
and of real functions that Pour-El and Richards have used in [6].
The following four deﬁnitions are introduced in [9] by Yasugi, Mori, and
Tsujii.
Deﬁnition 2.1 Let (M, d) be a metric space. A set S ⊂ Mω is a computability
structure on (M, d) if the following three conditions hold.
(i) If (xn), (yn) ∈ S, then (d(xn, yn′))n,n′ forms a computable double sequence
of real numbers.
(ii) If (xn) ∈ S and σ : N→ N is a recursive function, then (xσ(n)) ∈ S.
(iii) If (xn,k) ∈ S, (x′n) ∈ Mω, and (xn,k) converges to (x′n) eﬀectively in n
and k as k →∞, then (x′n) ∈ S.
An element of S is called a computable sequence in M .
Deﬁnition 2.2 A metric space with a computability structure (M, d,S) is
eﬀectively totally bounded if there exists a computable sequence (el) ∈ S and
a recursive function γ : N→ N such that
M = {el | l ∈ N} and (∀i ∈ N) M =
γ(i)⋃
l=0
B(el, 1/2
i).
H. Kamo / Electronic Notes in Theoretical Computer Science 120 (2005) 73–8274
(M, d,S) is eﬀectively compact if it is eﬀectively totally bounded and d is a
complete metric.
Deﬁnition 2.3 Let (M, d,S) be a metric space with a computability struc-
ture. A subset K ⊂ M is an eﬀectively compact subset of M if (K, d|K ,S∩Kω)
is an eﬀectively compact metric space.
In other words, K is an eﬀectively compact subset of (M, d,S) iﬀ it is a
compact subset of (M, d) and there exists a computable sequence (el) ∈ S and
a recursive function γ : N→ N such that
K = {el | l ∈ N} and (∀i ∈ N) K ⊂
γ(i)⋃
l=0
B(el, 1/2
i).
Deﬁnition 2.4 Let (M, d,S) be an eﬀectively compact metric space. A se-
quence of functions (fn), fn : M → R, is computable if the following two
conditions hold.
(i) (Sequential computability) For any computable sequence (xk) in M ,
(fn(xk))n,k forms a computable sequence of real numbers.
(ii) (Eﬀective uniform continuity) There exists a recursive function α :
N2 → N such that for any n, j ∈ N and any x, y ∈ M ,
d(x, y) < 1/2α(n,j) =⇒ |fn(x)− fn(y)| < 1/2j.
A function f : M → R is a computable function if (f)n∈N, the sequence all of
whose elements are equal to f , is a computable sequence of functions.
The recursive function α in Deﬁnition 2.4 (ii) is called an eﬀective modulus
of continuity of (fn).
3 Eﬀective Dini’s Theorem
First, we show two propositions with no assumptions on computability.
Proposition 3.1 Let (M, d) be a metric space. Let K ⊂ M be a nonempty
compact subset with (el) ∈ Mω and γ : N → N such that K = {el | l ∈ N}
and (∀i) K ⊂ ⋃γ(i)l=0 B(el, 1/2i). Then, for any ﬁnite sequence of open balls
(B(xk, rk))k=0,...,m, the following holds:
K ⊂
m⋃
k=0
B(xk, rk) ⇐⇒ (∃i)(∀l ≤ γ(i))(∃k ≤ m) d(xk, el) + 1
2i
< rk.
H. Kamo / Electronic Notes in Theoretical Computer Science 120 (2005) 73–82 75
Proof. (⇐=) Suppose y ∈ K. Then, (∃l ≤ γ(i)) d(y, el) < 1/2i. It follows
that
(∃i)(∃l ≤ γ(i))(∃k ≤ m)
[
d(y, el) <
1
2i
∧ d(xk, el) + 1
2i
< rk
]
.
This implies (∃k ≤ m) d(y, xk) < rk, i.e., y ∈
⋃m
k=0 B(xk, rk). This shows
K ⊂ ⋃mk=0 B(xk, rk).
(=⇒) We will prove the contraposition. The negation of the conclusion
is:
(∀i)(∃l ≤ γ(i))(∀k ≤ m) d(xk, el) + 1
2i
≥ rk.
By choosing an l for each i, we obtain a function γ ′ : N→ N such that
(∀i)(∀k ≤ m) d(xk, eγ′(i)) + 1
2i
≥ rk.
Since K is sequentially compact, there exists a subsequence (eγ′(θ(i))) that
converges to a point in K. Let y be the limit. A simple manipulation of
limits yields (∀k ≤ m) d(xk, y) ≥ rk, i.e., y ∈
⋃m
k=0 B(xk, rk). This shows
K ⊂ ⋃mk=0 B(xk, rk). 
Proposition 3.2 Let (M, d) be a separable metric space with a dense, at most
countable subset {el | l ∈ N}. Let a be a real number such that a ≥ 1. Let (εi)
be a sequence of positive real numbers converging to 0. Then,
B(x, r) =
⋃
l,i∈N,
d(x,el)+aεi<r
B(el, εi).
Proof. Comparison of the distance between the centers with the diﬀerence
between the radii yields B(x, r) ⊃ B(el, εi) if d(x, el) + aεi < r. This implies
B(x, r) ⊃ ⋃l,i∈N, d(x,el)+aεi<r B(el, εi).
Suppose y ∈ B(x, r). There exists an εi such that d(x, y) < r −
(a + 1)εi. Furthermore, there exists an el such that d(y, el) < εi. From
these two inequalities as well as d(x, el) ≤ d(x, y) + d(y, el), we obtain
d(x, el) + aεi < r. Therefore y ∈
⋃
l,i∈N, d(x,el)+aεi<r B(el, εi). This shows
B(x, r) ⊂ ⋃l,i∈N, d(x,el)+aεi<r B(el, εi). 
Next, we show two lemmata. Lemma 3.3 is on a consequence of eﬀective
compactness. Lemma 3.4 is on another characterization of computable real-
valued functions.
Lemma 3.3 Let (M, d,S) be a metric space with a computability structure.
For any eﬀectively compact subset K of M and any computable double sequence
H. Kamo / Electronic Notes in Theoretical Computer Science 120 (2005) 73–8276
(B(xn,k, rn,k)) of open balls in M , there exists a recursive partial function
α : N ⇀ N such that α(n) is deﬁned and K ⊂ ⋃α(n)k=0 B(xn,k, rn,k) holds if
K ⊂ ⋃∞k=0 B(xn,k, rn,k), and α(n) is undeﬁned otherwise.
Proof. We choose a computable sequence (el) and a recursive function γ :
N→ N such that K = {el | l ∈ N} and (∀i) K ⊂
⋃γ(i)
l=0 B(el, 1/2
i). By applying
Proposition 3.1 to each ﬁnite sequence (B(xn,k, rn,k))k=0,...,m for n = 0, 1, 2, . . . ,
we obtain
K ⊂
m⋃
k=0
B(xn,k, rn,k) ⇐⇒ (∃i)(∀l ≤ γ(i))(∃k ≤ m) d(xn,k, el) + 1
2i
< rn,k.
It is clear that the right-hand side is recursively enumerable in n and m.
Therefore, there exists a primitive recursive predicate ϕ on N3 such that
(∃i′)ϕ(n,m, i′) iﬀ K ⊂ ⋃mk=0 B(xn,k, rn,k). Hence, we can construct α by
α(n)  (min{m′ | ϕ(n, (m′)21, (m′)22)})21. 
Lemma 3.3 corresponds to “δ′range ≤ δHaine−Borel|K∗” shown by Brattka and
Presser in [1]. The proof here is essentially the same as that in [1] with some
correction for a minor error.
Lemma 3.4 Let (M, d,S) be an eﬀectively compact metric space. Let (el) ∈ S
be dense in M . For any sequence (fn) of real-valued functions on M , the
following two conditions are equivalent.
(i) (fn) is a computable sequence of functions.
(ii) There exists a recursively enumerable set S ⊂ N×Q×Q+ ×N×N such
that
(∀n ∈ N)(∀c ∈ Q)(∀r ∈ Q+) f−1n ((c− r, c + r)) =
⋃
(l,i)∈S(n,c,r)
B(el, 1/2
i).
Proof. [(i)⇒(ii)] Let α : N2 → N be an eﬀective modulus of continuity of
(fn). Let S ⊂ N×Q×Q+ × N×N be the set deﬁned by
(n, c, r, l, i) ∈ S
⇐⇒ (∃j ∈ N)(i = α(n, j) ∧ c− r < fn(el)− 1/2j ∧ fn(el) + 1/2j < c + r).
Since α is a recursive function and (fn(el)) is a computable double sequence
of real numbers, it follows immediately from the deﬁnition of S that S is a
recursively enumerable set.
H. Kamo / Electronic Notes in Theoretical Computer Science 120 (2005) 73–82 77
Let (l, i) ∈ S(n, c, r). Then we have, for some j ∈ N,
fn(B(el, 1/2
i)) = fn(B(el, 1/2
α(n,j)))
⊂ (fn(el)− 1/2j, fn(el) + 1/2j)
⊂ (c− r, c + r).
Therefore, B(el, 1/2
i) ⊂ f−1n ((c− r, c + r)). This shows f−1n ((c− r, c + r)) ⊇⋃
(l,i)∈S(n,c,r) B(el, 1/2
i).
Suppose x ∈ f−1n ((c− r, c + r)), i.e., |fn(x)− c| < r. Then, for some
j ∈ N, it holds that 2/2j ≤ r − |fn(x)− c|. For such a j, there exists an
el such that d(x, el) < 1/2
α(n,j). Hence |fn(el)− fn(x)| < 1/2j. By using
fn(el)− 1/2j < fn(x), we obtain
fn(el) + 1/2
j < fn(x) + 2/2
j ≤ fn(x) + r − |fn(x)− c| ≤ c + r.
Analogously, by using fn(el) + 1/2
j > fn(x), we obtain fn(el)− 1/2j > c− r.
The conjunction of the obtained two inequalities implies (n, c, r, l, α(n, j)) ∈
S. Therefore, x ∈ B(el, 1/2i) for some (l, i) ∈ S(n, c, r). This shows
f−1n ((c− r, c + r)) ⊂
⋃
(l,i)∈S(n,c,r) B(el, 1/2
i).
[(ii)⇒(i)] (Sequential computability) Suppose (xk) ∈ S. From (ii), we
obtain
|fn(xk)− c| < 1/2j ⇐⇒ (∃l)(∃i)[(n, c, 1/2j, l, i) ∈ S ∧ d(xk, el) < 1/2i].
Hence {(n, k, j, c) ∈ N × N × N × Q | |fn(xk)− c| < 1/2j} is a recursively
enumerable set. Meanwhile, (∀n)(∀k)(∀j)(∃c ∈ Q) |fn(xk)− c| < 1/2j holds
since Q is dense in R. Therefore, there exists a computable triple sequence of
rational numbers (cn,k,j) such that (∀n)(∀k)(∀j) |fn(xk)− cn,k,j| < 1/2j, i.e.,
(fn(xk))n,k is a computable double sequence of real numbers.
(Eﬀective uniform continuity) Using Proposition 3.2, we obtain that
B(el, 1/2
i) =
⋃
l′,i′∈N,
d(el,el′)+2/2i
′
<1/2i
B(el′ , 1/2
i′).
It is clear that d(el, el′) + 2/2
i′ < 1/2i is a recursively enumerable predicate
of l, i, l′, i′. It is also clear that (∀l)(∀i)(∃l′)(∃i′) d(el, el′) + 2/2i′ < 1/2i holds.
Hence there exist recursive functions σ, ρ : N3 → N such that for any l, i ∈ N,
{(l, i, l′, i′) ∈ N× N×N× N | d(el, el′) + 2/2i′ < 1/2i}
= {(l, i, σ(l, i, k), ρ(l, i, k)) | k ∈ N}.
H. Kamo / Electronic Notes in Theoretical Computer Science 120 (2005) 73–8278
Therefore, for any l, i ∈ N,
B(el, 1/2
i) =
∞⋃
k=0
B(eσ(l,i,k), 1/2
ρ(l,i,k)) (1)
and
(∀k) d(el, eσ(l,i,k)) + 2/2ρ(l,i,k) < 1/2i. (2)
Since S is a recursively enumerable set, so is {(n, j, l, i, c) ∈ N × N ×
N × N × Q | (n, c, 1/2j+1, l, i) ∈ S}. On the other hand, since M =⋃
c∈Q f
−1
n ((c− 1/2j+1, c + 1/2j+1)), it holds that
M =
⋃
c∈Q
⋃
(l,i)∈S(n,c,1/2j+1)
B(el, 1/2
i).
Hence (∀n)(∀j)(∃l)(∃i)(∃c ∈ Q) (n, c, 1/2j+1, l, i) ∈ S. Therefore, there exist
recursive functions σ′, ρ′ : N3 → N and a computable triple sequence of rational
numbers (cn,j,k) such that for any n, j ∈ N,
{(n, j, l, i, c) ∈ N× N× N×N×Q | (n, c, 1/2j+1, l, i) ∈ S}
= {(n, j, σ′(n, j, k), ρ′(n, j, k), cn,j,k) | k ∈ N}.
Thus, for any n, j ∈ N,
M =
∞⋃
k=0
B(eσ′(n,j,k), 1/2
ρ′(n,j,k)) (3)
and
(∀k) fn(B(eσ′(n,j,k), 1/2ρ′(n,j,k))) ⊂ (cn,j,k − 1/2j+1, cn,j,k + 1/2j+1). (4)
From (1) and (3), we obtain
M =
∞⋃
k=0
B(eσ(σ′(n,j,(k)21),ρ′(n,j,(k)21),(k)22), 1/2
ρ(σ′(n,j,(k)21),ρ
′(n,j,(k)21),(k)
2
2)).
Application of Lemma 3.3 yields that there exists a recursive function γ :
N2 → N such that
M =
γ(n,j)⋃
k=0
B(eσ(σ′(n,j,(k)21),ρ′(n,j,(k)21),(k)22), 1/2
ρ(σ′(n,j,(k)21),ρ
′(n,j,(k)21),(k)
2
2)).
H. Kamo / Electronic Notes in Theoretical Computer Science 120 (2005) 73–82 79
Let α : N2 → N be a recursive function deﬁned by
α(n, j) = max
k≤γ(n,j)
ρ(σ′(n, j, (k)21), ρ
′(n, j, (k)21), (k)
2
2).
Suppose points x, y ∈ M satisfy d(x, y) < 1/2α(n,j). Then there exists some
k ≤ γ(n, j) such that
d(x, eσ(σ′(n,j,(k)21),ρ′(n,j,(k)21),(k)22)) < 1/2
ρ(σ′(n,j,(k)21),ρ
′(n,j,(k)21),(k)
2
2).
With such a k,
d(x, eσ′(n,j,(k)21))
≤ d(eσ′(n,j,(k)21), eσ(σ′(n,j,(k)21),ρ′(n,j,(k)21),(k)22)) + d(x, eσ(σ′(n,j,(k)21),ρ′(n,j,(k)21),(k)22))
< d(eσ′(n,j,(k)21), eσ(σ′(n,j,(k)21),ρ′(n,j,(k)21),(k)22)) + 1/2
ρ(σ′(n,j,(k)21),ρ
′(n,j,(k)21),(k)
2
2)
< 1/2ρ
′(n,j,(k)21) − 1/2ρ(σ′(n,j,(k)21),ρ′(n,j,(k)21),(k)22).
Furthermore,
d(y, eσ′(n,j,(k)21)) ≤ d(x, eσ′(n,j,(k)21)) + d(x, y)
< 1/2ρ
′(n,j,(k)21) − 1/2ρ(σ′(n,j,(k)21),ρ′(n,j,(k)21),(k)22) + 1/2α(n,j)
≤ 1/2ρ′(n,j,(k)21).
Therefore, x, y ∈ B(eσ′(n,j,(k)21), 1/2ρ
′(n,j,(k)21)). Due to (4), this implies
fn(x), fn(y) ∈ (cn,j,(k)21 − 1/2j+1, cn,j,(k)21 + 1/2j+1). Hence |f(x)− f(y)| <
1/2j. This shows that α is an eﬀective modulus of continuity of (fn). 
The condition (ii) in Lemma 3.4 is equivalent to δ6-computability deﬁned
by Weihrauch in [7]. Lemma 3.4 shows that for a real-valued function on an
eﬀectively compact metric space, computability deﬁned by Mori, Tsujii, and
Yasugi in [5] and used in this paper is equivalent to δ6-computability.
Now we are ready to show the main theorem.
Theorem 3.5 Let (M, d,S) be an eﬀectively compact metric space. Let (fn)
be a computable sequence of real-valued functions on M and f a computable
real-valued function on M . If fn converges pointwise monotonically to f as
n →∞, then fn converges eﬀectively uniformly to f .
Proof. Let (el) be a computable sequence dense in M .
Let Uj,n = (fn− f)−1((−1/2j, 1/2j)). Since fn converges pointwise to f , it
holds that (∀j) M = ⋃∞n=0 Uj,n. Due to Lemma 3.4, there exists a recursively
H. Kamo / Electronic Notes in Theoretical Computer Science 120 (2005) 73–8280
enumerable set S ⊂ N4 such that (∀j)(∀n) Uj,n =
⋃
(i,l)∈S(j,n) B(el, 1/2
i). From
these two equalities, we obtain
(∀j) M =
⋃
(n,i,l)∈S(j)
B(el, 1/2
i).
This implies (∀j) S(j) = ∅. Therefore, there exist recursive functions θ, ρ, σ :
N2 → N such that S = {(j, θ(j, k), ρ(j, k), σ(j, k)) | j, k ∈ N}. Using these
functions, we can rewrite the formula above as follows:
(∀j) M =
∞⋃
k=0
B(eσ(j,k), 1/2
ρ(j,k)).
Since M itself is a compact subset of M , application of Lemma 3.3 yields that
there exists a recursive total function β : N→ N such that
(∀j) M =
β(j)⋃
k=0
B(eσ(j,k), 1/2
ρ(j,k)).
Since B(eσ(j,k), 1/2
ρ(j,k)) ⊂ Uj,θ(j,k), this implies (∀j) M =
⋃β(j)
k=0 Uj,θ(j,k). Since
fn converges monotonically to f , it holds that Uj,n ⊂ Uj,n′ if n ≤ n′. Let
α : N → N be a recursive function deﬁned by α(j) = maxk≤β(j) θ(j, k). We
have (∀j)(∀n ≥ α(j)) M = Uj,n, which is equivalent to:
(∀j)(∀n ≥ α(j))(∀x ∈ M) |fn(x)− f(x)| < 1/2j.
Thus fn converges eﬀectively uniformly to f as n →∞. 
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Effective Dini's Theorem on Effectively Compact Metric Spaces

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