Effective zero-dimensionality for computable metric spaces

0.34 MB·English

by  Robert Kenny

LogicalMethodsinComputerScience
Vol.11(1:11)2015,pp.1–25 Submitted Nov. 9,2013
www.lmcs-online.org Published Mar.25,2015
EFFECTIVE ZERO-DIMENSIONALITY FOR COMPUTABLE
METRIC SPACES∗
ROBERTKENNY
School of Mathematics & Statistics, The University of Western Australia, Perth, Australia
e-mail address: [email protected]
Abstract. We begin to study classical dimension theory from the computable analy-
sis (TTE) point of view. For computable metric spaces, several effectivisations of zero-
dimensionalityareshowntobeequivalent. Thepartofthischaracterisationthatconcerns
covering dimension extends to higher dimensions and to closed shrinkings of finite open
covers. To deal with zero-dimensional subspaces uniformly, four operations (relative to
the space and a class of subspaces) are defined; these correspond to definitions of induc-
tiveand coveringdimensionsandacountablebasiscondition. Finally,an effectiveretract
characterisationofzero-dimensionalityisprovenunderaneffectivecompactnesscondition.
In one direction this uses a version of theconstruction of bilocated sets.
1. Introduction
Variousspacesofsymbolicdynamics[12],suchasX = AN forafinitealphabetAorthesofic
subshifts, are useful examples of zero-dimensional topological spaces, interesting both for
dynamics and in connection with computation. Some similar remarks apply to the spaces
of cellular automata AZn and to a lesser extent to general subshifts. To deal effectively with
sets which are zero-dimensional in non-symbolic mathematical contexts, however (such as
in Rn or the minimal sets of an expansive compact dynamical system [1, Thm 2.2.44]), it
is desirable to examine possible effective versions of this property. In the present work,
we begin a basic investigation to consider effective zero-dimensionality both of computable
metric spaces and of their closed subsets, in the framework of computable analysis via
representations (see [18], [8]).
To this end, for a topological space X, recall that a subset B ⊆X is clopen if B is open
and closed, equivalently if the boundary ∂B is empty. For a separable metrizable space X,
the following conditions are equivalent:
(1) (∀p ∈X)(∀A ∈ Π0(X))(p 6∈ A =⇒ ∅ is a partition between p and A),
1
(2) (∀A,B ∈ Π0(X))(A∩B = ∅ =⇒ ∅ is a partition between A and B),
1
(3) (∀U)(∃V)(U is an open cover of X =⇒ V is a partition into open sets refining U),
2012 ACM CCS: [Mathematics of computing]: Continuous mathematics—Topology—Point-set
topology.
Key words and phrases: zero-dimensional, dimension theory,computable analysis.
∗ Resultsin Section 4 havebeen presented at CCA 2013, Nancy,France, Jul 8-10 2013.
LOGICALMETHODS (cid:13)c R.Kenny
lINCOMPUTERSCIENCE DOI:10.2168/LMCS-11(1:11)2015 (cid:13)CC CreativeCommons
2 R.KENNY
(4) (∀U)(∃V) U is a finite open cover of X =⇒ V is a finite partition into open sets
refining U ,
(cid:0)
(5) there exists a countable basis B for the topology of X consisting of clopen sets,
(cid:1)
(6) (∀U)(∃V) U is a finite open cover =⇒ V is an open shrinking of U by pairwise
disjoint sets ,
(cid:0)
(7) (∀A∈ Π0(X))(∃f ∈ C(X,X))(A 6= ∅ =⇒ imf = A∧f| = id ).
1 (cid:1) A A
Here, in (1) and (2), P is a partition between disjoint A,B ⊆ X if there exist disjoint open
U,V ⊆ X such that A ⊆ U, B ⊆ V and X \P = U ∪V. In (3) and (4) a partition (of
X) is a pairwise disjoint family of sets (with union equal to X). In (6) a shrinking of a
cover (A ) of X is a cover (B ) satisfying B ⊆ A for all i ∈ I. A nonempty space X
i i∈I i i∈I i i
satisfying (1) (or any of the equivalent conditions) is zero-dimensional; a subset Y ⊆ X is
zero-dimensional if Y is zero-dimensional in the relative topology T |Y.
X
Next, recall that any zero-dimensional separable metrizable X is homeomorphic to a
subspace of the Cantor space C := {0,1}N [10, Thm 7.8, p 38]. For strictly topological
questions on zero-dimensional spaces it is thus possible to consider only subspaces of C.
In this paper we will address our questions from the slightly more intrinsic point of view
mentioned above, treating zero-dimensionality on a computable metric space X and its
subsets. More specifically, we consider computable versions of the existence statements (1)-
(7); these are certain multi-valued operations which, stopping short of studying Weihrauch
degrees, we require to be computable. In the case of a subset Y ⊆ X, zero-dimensionality
of Y can be stated in several ways using closed or open subsets of X, and these statements
also can be viewed as multi-valued operations. While a systematic treatment is not given,
we present various definitions of operations (corresponding to equivalent forms of zero-
dimensionality) and some of their interrelations.
Thus, in Section 4 three implications are proven between four operations relevant for a
general class Y ⊆ P(X) of zero-dimensional or empty subsets of cardinality |Y| ≤ 2ℵ0; these
correspond to (1), (2), (5) and, loosely, to a condition like (3) or (6). Further results on
the four operations for Y = {Y ∈ Π0(X) | dimY ≤ 0} under effective local compactness or
1
similarassumptionswillbediscussedelsewhere. InSection5theresultsofSection 4arespe-
cialised to the case Y = X, and a robust notion of effectively zero-dimensional computable
metric space is found to exist. Some more evidence for the suitability of that definition
is provided by Section 6, which deals with covering dimension (essentially extending the
conditions (3), (4) and (6)), though in an ad hoc way.
We also present, in Section 7, an effective version of the decomposition of totally
bounded open subsets of zero-dimensional spaces found in [11, Cor 26.II.1]. This is used
(with an effective compactness assumption) to prove Theorem 7.6, an effectivization of (7)
above. Finally, in Section 8 a converse Proposition 8.5 is proven. This relies on the exis-
tence of so-called bilocated sets from the constructive analysis literature; some computable
analysis versions of these proofs are given in the same section. Sections 2 and 3 respectively
discuss notation and supporting results on general covering properties of metric spaces
(namely, effective versions of the Lindel¨of property, and swelling and shrinking of finite
covers).
EFFECTIVE ZERO-DIMENSIONALITY 3
2. Notation
By h·i : N∗ → N and h·,·i : N2 → N we denote standard tupling functions, with corre-
sponding coordinate projections π ,π : N → N in the binary case. A standard numbering
1 2
ν of N∗ is also introduced by ν hwi := w (w ∈ N∗). Similarly, with B := NN we define
N∗ N∗
h·,·i :B2 → B and h·,...i: BN → B by
hp(0),p(1)i(2i+z)= p(z) and hp(0),p(1),...i(hi,ji) = p(i)
i j
(here p = p p ··· ∈ B, i.e. p := p(i) for every p ∈ B, i ∈ N). Again we write π ,π :
0 1 i 1 2
B → B for the coordinate projections in the binary case. We will also occasionally consider
projections π : X ×Y → X and π : X ×Y → Y for any cartesian product X ×Y; it will
1 2
be clear from the context which of the above notions is meant. Further, in a metric space
X, we write
N (A) := B(x;ǫ) and N¯ (A) := B¯(x;ǫ)
ǫ ǫ
x∈A x∈A
[ [
for any A⊆ X and ǫ > 0.
In general, we assume familiarity with the framework of computable analysis via rep-
resentations [18], [8]. We will also use some notation for specific representations from [5].
If (X ,δ ) (1 ≤ i ≤ n) and (Y,δ′) are represented spaces, similarly to [18], a (δ ,...,δ ;δ′)-
i i 1 n
realiser of an operation f :⊆ X ×···×X ⇒ Y is a map F :⊆ Bn → B such that
1 n
F(p(1),...,p(n))∈ (δ′)−1f(δ (p(1)),...,δ (p(n))) whenever
1 n
n
(p(1),...,p(n)) ∈ domδ and (δ (p(1)),...,δ (p(n))) ∈ domf.
i 1 n
i=1
Y
However, unless otherwise mentioned, when representations δ , δ , δ′ are understood a
1 2
‘realiser’ of f :⊆ X ×X ⇒ Y will be a map F :⊆ B → B, namely a ([δ ,δ ];δ′)-realiser.
1 2 1 2
This convention has some minor advantages where brevity is concerned.
For a computable metric space (X,d,ν), in this paper the Cauchy representation δ :⊆
X
B → X is defined by
p ∈ δ−1{x} :⇐⇒ lim ν(p )= x∧(∀i,j ∈ N)d(ν(p ),ν(p )) < 2−min{i,j}.
X i i j
i→∞
A representation ρ of R will be used less often; for definiteness, let it be the Cauchy repre-
sentation of (R,d,ν ), where d(x,y) = |x−y|.
Q
Let (X,T)bea second countable topological space andlet α,β : N→ T benumberings
of possibly different countable bases.
Definition 2.1. (⊏) ⊆ N2 is a formal inclusion of α with respect to β if
(∀a,b ∈ N)(a ⊏ b =⇒ α(a) ⊆ β(b)).
Consider the following axioms, in order of increasing strength.
(1) (∀b)(∀x ∈ X)(∃a)(x ∈ β(b) =⇒ x ∈ α(a)∧a ⊏ b)
(2) (∀b)(∀x ∈ X)(∀U ∈ T)(∃a)(x ∈β(b)∩U =⇒ x ∈ α(a) ⊆ U ∧a ⊏ b)
(3) (∀a,b)(∀x ∈ X)(∃c)(x ∈ β(a)∩β(b) =⇒ x ∈ α(c)∧c ⊏ a∧c⊏ b)
(4) (∀b)(∀x ∈ X)(∃U ∈ Σ0(X))(∀a)(x ∈ α(a) ⊆ β(b)∩U =⇒ a ⊏ b)
1
4 R.KENNY
In particular, in a computable metric space (X,d,ν), consider numberingsof ideal open
and closed balls
α :N → B := imα ⊆ T,ha,ri 7→ B (ν(a);ν (r)),
d Q+
αˆ :N → imαˆ ⊆ Π0(X),ha,ri 7→ B¯ (ν(a);ν (r)).
1 d Q+
Here ν is a standard total numbering of the positive rationals Q+ with a (ν ,id )-
Q+ Q+ N
computable right-inverse · :Q+ → N.
The relation ⊏ defined by
ha,ri ⊏ hb,qi :⇐⇒ d(ν(a),ν(b))+ν (r) < ν (q)
Q+ Q+
is a formal inclusion of α with respect to itself; moreover it satisfies c ⊏ d =⇒ αˆ(c) ⊆ α(d)
and (4). For the purposes of this paper, we will often call a formal inclusion satisfying
property (1) a refined inclusion.
From any basis numbering α (of a topological space X) we can define a representation
δ : B → Σ0(X),p 7→ {α(p −1) |i ∈ N,p ≥ 1}
1 i i
of the hyperspace of open sets in X. For[a computable metric space with α as above, this
representation is denoted δ , or δ if X is clear from the context. Correspondingly, we
Σ0(X) Σ0
1 1
write
δ :B → Π0(X),p 7→ X \δ (p)
Π0 1 Σ0
1 1
for a representation of the hyperspace of closed sets in X, and
δ :⊆ B→ ∆0(X),hp,qi 7→ δ (p)= δ (q)
∆0 1 Σ0 Π0
1 1 1
(with natural domain) for a representation of the clopen sets in X. When writing Σ0(X),
1
Π0(X), ∆0(X) we always assume these classes are equipped with the corresponding repre-
1 1
sentations.
For the purposes of this paper we need two more representations of the class A(X) of
closed sets in X (cf. [5]). Define δ ,δ> :⊆ B → A(X) by
range dist
hp(0),...i ∈ δ−1 {A} : ⇐⇒ A = ∅∧(∀i)p(i) =0ω ∨ A 6= ∅∧{p(i) | i ∈N} ⊆ P−1δ−1A∧
range X
(cid:0)(∀x ∈ A)(∀U ∈ TX)(∃(cid:1)i)((cid:0)x ∈ U =⇒ (δX ◦P)(p(i)) ∈ U) ,
where P :⊆ B → B is defined by P(p)i := pi−1 (domP = {p ∈B | (∀i)pi ≥ 1}), (cid:1)
p ∈ (δ> )−1{A} :⇐⇒ η (δ ,ρ )-realises d :X → R¯,
dist p X < A
where
p ∈ ρ −1{t} :⇐⇒ {n ∈N | ν (n)< t} = {p −1 |i ∈ N∧p ≥ 1}.
< Q i i
Here η = η(p) for a certain ‘canonical’ representation η of the set F = {F :⊆ B → B |
p
F continuous with G domain}. In particular, η satisfies certain versions of the smn and
δ
utm theorems; see [18, §2.3] for precise (and rather general) statements. δ and δ> will
range dist
be used in Sections 7 and 8.
Next, for any represented set (X,δ), consider the set X∗ of finite-length words over the
alphabet X. A representation of X∗ is defined by
λ, if n = 0
δ∗ :⊆ B → X∗,n.hp(0),p(1),...i7→ (n ∈ N),
(δ(p(0))...δ(p(n−1)), if n ≥ 1
EFFECTIVE ZERO-DIMENSIONALITY 5
whereλistheemptyword. InSections3,5,6and8wewilluseδ∗ forvariousrepresentations
δ of hyperspaces of a fixed computable metric space X. If (I,ν) is a numbered set, a
representation δ :⊆ B → I is defined by
ν
domδ = {p ∈B | p ∈ domν} and δ (p) = ν(p ).
ν 0 ν 0
Consider now the set E(X) of finite subsets of X. For a numbered set (I,ν) one can define
a standard numbering FS(ν) of E(I) following [16, Defns 2.2.2, 2.2.14(5)]: first, define a
total numbering e of E(N) by e = ψ−1 for the bijection ψ :E(N) → N,A 7→ 2i. Then
i∈A
define
P
FS(ν) :⊆ N → E(I),k 7→ {ν(i) |i ∈ e(k)} where domFS(ν) = {k |e(k) ⊆ domν}.
The next lemma verifies equivalence of two representations arising from these definitions.
Lemma 2.2. For any numbered set (I,ν), δ ≡ δ , where
FS(ν) E(I)
p ∈ δ−1 {S} : ⇐⇒ (∃k)(∀i)((i < k =⇒ p ∈ 1+domν)∧(i≥ k =⇒ p = 0))
E(I) i i
∧{ν(p −1) | i< k} = S.
i
Proof. δFS(ν) ≤ δE(I): we use F :⊆ B → B,a.0ω 7→ w.0ω where |w| = #e(a) (the number
of nonzero bits in the binary representation of a) and w := j +1 if j is the ith smallest
i
member of e(a).
δE(I) ≤ δFS(ν): we use F :⊆ B → B,p 7→ a.0ω where k := µi(pi = 0) and a := {2j | j ∈
N∧(∃i< k)p = j +1}.
i
P
3. Covering properties
For any represented spaces (X,δ), (Y,δ′), denote the set of (δ,δ′)-continuous total maps
f :X → Y by C (δ,δ′).
s
Lemma 3.1. For computable metric spaces (X,d,ν), (Z,d′,ν′) and Cauchy representation
δ of Z, the computable dense sequence z := ν′(i) (i ∈ N) satisfies
Z i
u(z ) = u(z)
i∈N i z∈Z
for any u ∈ C (δ ,δ ). In particular,
s Z Σ0(X) S S
1
L′ :C (δ ,δ ) → Σ0(X)N,u7→ (u(z )) ,
s Z Σ0(X) 1 i i∈N
1
∪ :C (δ ,δ ) → Σ0(X),u 7→ u(z)
s Z Σ0(X) 1 z∈Z
1
are resp. ([δZ → δΣ01(X)],δΣω01(X))- and ([δZ → δΣ01(X)],δΣ01S(X))-computable.
Lemma3.1playsasimilarroletotheLindel¨ofpropertyofseparablemetricspaces,albeit
only for representation-continuous indexed covers. Theoperation of continuous intersection
for closed subsets, dual to ∪, has been considered in [6].
Proof. Take
A:= {w ∈ N∗ | w.B∩domδ 6= ∅} = {w ∈ N∗ | (∀i,j < |w|)d′(ν′(w ),ν′(w )) < 2−min{i,j}}.
Z i j
6 R.KENNY
We let z := ν′(i) = ν′(w ) for λ 6= w ⊏ iω. Now consider u ∈ C (δ ,δ ) and a
i |w|−1 s Z Σ0(X)
1
continuous realiser F :⊆ B → B of u. For any x ∈ X and z ∈ Z such that u(z) ∋ x and
q ∈ δ−1{z}, it holds that
Z
u(z) = (δ ◦F)(q) = {α(F(q) −1) |n ∈ N,F(q) ≥ 1};
Σ0(X) n n
1
we suppose x ∈ α(a) where a+1 = F(q[) . Since F is continuous, there exists w ⊏ q such
n
that any r ∈ w.B∩domδ satisfies F(r) = a+1 and hence (u◦δ )(r) ∋ x. In particular
Z n Z
this applies to r = w.wω ∈ δ−1{ν′(w )} = δ−1{z } for i= w .
|w|−1 Z |w|−1 Z i |w|−1
We continue this section with some results around shrinkings and swellings of covers;
as in the classical case these are useful to give equivalent definitions of bounds on covering
dimension. Following [9], these constructions depend on Urysohn’s lemma; we specifically
are interested in the effective form from [17].
Theorem 3.2. (Weihrauch [17, Thm 15]) In a computable metric space X, define
U :⊆ Π0(X)2 ⇒ C(X,R),(A,B) 7→ {f | imf ⊆ [0,1]∧f−1{0} =A∧f−1{1} = B}
1
(domU = {(A,B) | A∩B = ∅}). Then U is ([δ ,δ ],[δ → ρ])-computable.
Π0 Π0 X
1 1
Definition 3.3. For any family A = (A ) ⊆ P(X), a swelling of A is a family (B )
i i∈I i i∈I
satisfying (∀i)(A ⊆ B ) and
i i
B = ∅ ⇐⇒ A = ∅ (3.1)
wj wj
j<m j<m
\ \
for any m ≥ 1, w ∈ Im.
Classically, anyfinitecollection of closed subsetshasanopenswellingandthisconstruc-
tioncanbeeffectivized givensuitabledataontheemptinessornonemptinessofintersections
in (3.1). Dually, this result allows (δ∗ - and) subcover information for a finite open cover to
Σ0
1
be used to produce closed or open shrinkings computably. For the present paper, working
withsuchinformation(codingitappropriatelyinrepresentationsforcovers) isunnecessarily
complicated; we instead consider two partial effectivisations of the proof of [9, Thm 7.1.4].
For any indexed family (A ) ⊆ P(X), the order of the family, ord(A ) , is here defined
i i∈I i i∈I
as the least n such that A is empty whenever i ,...,i are distinct elements of I
j≤n ij 0 n
(this definition varies slightly from that in [9]).
T
Lemma 3.4. Let X be a computable metric space. For any N ∈ N, the operation S :⊆
+,N
Π0(X)∗ ⇒ Σ0(X)∗ defined by domS = {(F ) | (F ) of order ≤ N +1} and
1 1 +,N i i<k i i
S ((F ) )= {(U ) |(∀i)(F ⊆ U ) and (U ) of order ≤ N +1}
+,N i i<k i i<k i i i i<k
is (δ∗ ,δ∗ )-computable.
Π0 Σ0
1 1
Proof. Assuming (F ) ∈ domS , we first deal with the case k ≥ N +2. Inductively
i i<k +,N
in n < k, assume f ∈ C(X,[0,1]) has F ⊆ f−1{0} and K := f−1[0,2−1] for each i < n.
i i i i i
(n) (n)
We also assume (F ) is of order at most N + 1 and F ⊆ F for all i < k where
i i<k i i
(n)
F := (K , if i < n;F , if n ≤ i < k) (i< k). Then
i i i
S := { F(n) |w ∈ [0,k)N+2 injective with w = n}
n wj N+1
j≤N
[ \
EFFECTIVE ZERO-DIMENSIONALITY 7
(n)
is closed and disjoint from F = F . By Urysohn’s lemma there exists continuous f :
n n n
X → [0,1] suchthatF ⊆ f−1{0} andS ⊆ f−1{1}. DefiningK and(F(n+1)) asabove,
n n n n n i i<k
we have F(n) ⊆ F(n+1) for each i < k, and for w ∈ [0,k)∗ injective with |w| ≥ N +2 we
i i
have
(n)
F = ∅, if (∀j < |w|)(w 6= n)
F(n+1) = j<|w| wj j ⊆ K ∩S = ∅.
j<|w| wj (KTn∩ j6=j′<|w|Fw(nj′), if wj = n n n
\
T (n+1)
Instepn = k−1oftheaboveinductionwegetF = K foralli< kandanyinjective
i i
w ∈ [0,k)∗ with |w| ≥ N +2 satisfies K = ∅. But then for U := f−1[0,2−1) ⊆ K
j<|w| wj i i i
(i < k) it is clear (U ) ∈ S ((F ) ). Furthermore it is clear how to obtain δ∗ -
i i<k +,N Ti i<k Σ0
1
informationon(U ) . Namely, letF andGbefixedcomputablerealisersof theoperations
i i<k
T(N) :⊆ Π0(X)∗ ×C(X,R)∗ → Π0(X),((F ) ,(f ) ) 7→ S
1 1 i i<k i i<n n
(domT(N) = {((F ) ,(f ) ) | n ≤ k}) and U :⊆ Π0(X)2 ⇒ C(X,R) (from Theorem
i i<k i i<n 1
3.2), and p = k.hp(0),...,p(k−1),0ω,0ω,...i ∈(δ∗ )−1{(F ) }. Then
Σ0 i i<k
1
q(n) := Ghp(n),Fhp,n.hq(0),...,q(n−1),0ω,0ω,...iii (0 ≤ n < k)
are [δ → ρ]-names of respective f , uniformly computable from the inputs; computability
X n
here is a matter of appropriate dovetailing. Note for the case k ≤ N+1 the same argument
works (with S = ∅ for all i < k); in any case, checking (U ) have order at most N +1
i i i<k
becomes trivial. This completes the proof.
Proposition 3.5. For any computable metric space X, the operations
S : ⊆ Σ0(X)∗ ⇒ Π0(X)∗,(U ) 7→ {(F ) | (∀i)(F ⊆ U )∧ F = X}
− 1 1 i i<k i i<k i i i i
T : ⊆ Π0(X)∗ ⇒ Σ0(X)∗,(B ) 7→ {(U ) | U = ∅∧(∀i)B ⊆ U }
1 1 i i<k i i<k i i Si i
(domS = {(U ) | U = X}, domT = {(B ) | B = ∅}) are resp. (δ∗ ,δ∗ )- and
− i i<k i i i i<k Ti i Σ0 Π0
1 1
(δ∗ ,δ∗ )-computable.
Π0 Σ0 S T
1 1
Proof. Inductively in n ≤ k, suppose (B(n)) , f ∈ C(X,[0,1]) and K := f−1[0,2−1]
i i<k i i i
(n)
(i < n) are such that B = (K , if i < n; B , if n ≤ i< k). We additionally suppose that
i i i
f satisfy B ⊆ f−1{0} for all i< n, and that B(n) = ∅.
i i i i<k i
(n) (n)
Then S := B is closed and disjoint from B (= B ). By Urysohn’s
n j∈[0,k)\{n} j T n n
lemma, there exists continuous f : X → [0,1] such that B ⊆ f−1{0} and S ⊆ f−1{1}.
T n n n n n
(n+1) (n) (n+1)
Defining K , (B ) as above in the case n+1 ≤ k, we have B ⊆ B for all
n i i<k i i
i < k with
(n+1) (n)
B = K ∩ B = B ∩K = S ∩K = ∅.
i<k i i≤n i n<i<k i n6=i<k i n n n
By stTep k of this indTuction, theTre exist fi suTch that Bi ⊆ fi−1{0} and Ki := fi−1[0,2−1]
(i < k) satisfy B ⊆ K for all i and K = ∅. Writing U := f−1[0,2−1) (i < k) we now
i i i i i i
have B ⊆ U ⊆ K for all i < k and U = ∅. This establishes the computability of T.
i i i i i
T
Then domS = {(X \ B ) | (B ) ∈ domT}, and also (F ) ∈ S ((U ) ) iff (X \ F ) ∈
− i i i i i i − i i i i
T
T((X \U ) ).
i i
8 R.KENNY
4. Zero dimensional subsets
For a computable metric space X and a class Y ⊆ P(X) of zero-dimensional or empty sub-
sets with |Y| ≤ 2ℵ0, what information should be included (or more abstract requirements
made) when specifying a representation δ of Y? Loosely speaking, we would like effective
Y
versions of certain theorems concerning zero-dimensionality to hold, without requiring ‘un-
realistically’ strong information on inputs. While we are here far from an exposition that
would satisfactorily answer this open-ended problem, it seems a reasonable place to start is
from thedefinition of zero-dimensionality as presented in Section 1. Specifically, as effectivi-
sations of (5), (6), (1), (2) which also depend on the subspace Y in place of X we (for given
X, Y, δ ) consider computability of respective operations B, S, M, N, defined as below.
Y
For brevity, in case of the binary disjoint union of two sets, we often write “E = C∪˙D” in
place of “C ∩D = ∅ and E = C ∪D”.
B :Y ⇒ (Σ0(X)2)N ×(N2)N, S :⊆ Σ0(X)N ×Y ⇒ Σ0(X)N,
1 1 1
M :⊆ X ×Σ0(X)×Y ⇒ Σ0(X)2, N :⊆ Π0(X)2 ×Y ⇒ Σ0(X)2
1 1 1 1
with domS = {((U ) ,Y) | U ⊇ Y}, domM = {(x,U,Y) | x ∈ U}, domN =
i i i i
{(A,B,Y) |A∩B = ∅}, and
S
B(Y) := ((U ,V ) ,(a ,b ) ) |(U ) a basis for T , (∀i)Y ⊆ U ∪˙V and
i i i i i i i i X i i
{(ak(cid:8),bk)| k ∈ N} refined inclusion of (Ui)i w.r.t. α ,
S((V ) ,Y) := {(W ) | (∀i)W ⊆ V ∧ W ⊇ Y and (W ) pairwise disjoint},
i i i i i i i i i i (cid:9)
M(x,U,Y) := {(V,W) | x ∈ V ⊆ U ∧Y ⊆ V∪˙W},
S
N(A,B,Y) := {(U,V) |A ⊆ U ∧B ⊆ V ∧Y ⊆ U∪˙V}
Proposition 4.1. Let X be a computable metric space and Y ⊆ P(X) a class of zero-
dimensional or empty subsets with representation δ . Then ((i)) =⇒ ((ii)) =⇒ ((iii))
Y
=⇒ ((iv)).
(i) N :⊆ Π0(X)2 ×Y ⇒ Σ0(X)2 is computable.
1 1
(ii) M :⊆ X ×Σ0(X)×Y ⇒ Σ0(X)2 is computable.
1 1
(iii) B : Y ⇒ (Σ0(X)2)N ×(N2)N is computable.
1
(iv) S :⊆ Σ0(X)N ×Y ⇒ Σ0(X)N is computable.
1 1
Proof.
((i)) =⇒ ((ii)): If (x,U,Y) ∈ domM then ({x},X \U,Y) ∈ domN and for any (V,W) ∈
N({x},X \ U,Y) it holds that x ∈ V ⊆ X \ W ⊆ U and Y ⊆ V ∪ W (equivalently,
(V,W) ∈ M(x,U,Y) and X \U ⊆ W).
((ii)) =⇒((iii)): Consider M◦ : X ×N×Y ⇒ Σ0(X)2 defined by
1
M◦(x,i,Y) = M(x,B(x;2−i),Y) = {(V,W) | x ∈ V ⊆ B(x;2−i)∧Y ⊆ V∪˙W}.
If δ = δ is Cauchy representation of X, let G :⊆ B3 → B be a computable (δ,δ ,δ ;δ2 )-
X N Y Σ0
1
realiser of M◦, Z := domδ and
u(j) := (δ ◦π ◦G)(·,j.0ω,q) : Z → Σ0(X) (j ∈ N, q ∈ domδ ).
q Σ0 1 1 Y
1
We now can apply Lemma 3.1 to u(j) (with (p(i)) a standard enumeration of {w.wω |
q i |w|−1
w ∈ A} ⊆Z for A as in proof of the lemma), obtaining u(j)(p(i)) = X for every j ∈ N,
i∈N q
S
EFFECTIVE ZERO-DIMENSIONALITY 9
q ∈ domδ . If we denote
Y
B :N×B → Σ0(X)2,(hi,ji,q) 7→ (δ2 ◦G)(p(i),j.0ω,q)
1 1 Σ0
1
and b′ := π ◦B : N×B→ Σ0(X) then one can check each b′(·,q) is a basis numbering.
1 1 1
Next, define
hi,ji ⊏′ hn,ri: ⇐⇒ d(δ(p(i)),ν(n))+2−j < ν (r).
Q+
We show
Property 1. (⊏′) ⊆ N2 is a c.e. refined inclusion of b′(·,q) w.r.t. α.
Proof of Property 1:
b′(hi,ji,q) = u(j)(p(i)) = (π ◦δ2 ◦G)(p(i),j.0ω,q) ∈ (π ◦M◦)(δ(p(i)),j,δ (q))
q 1 Σ0 1 Y
1
implies δ(p(i)) ∈ b′(hi,ji,q) ⊆ B(δ(p(i));2−j), where the latter set is included in αhn,ri if
hi,ji ⊏′ hn,ri.
Secondly, for p ∈ B and N ∈ N let pN denote the prefix p ...p of p. We let s ∈ N,
0 N−1
y := ν(π s), r := ν (π s), x ∈ α(s), p ∈ δ−1{x} and define
1 Q+ 2
H(j) :⊆ B → B,p 7→ (π ◦G)(p,j.0ω,q).
q 1
H(j) is a continuous (id |Z,δ )-realiser of u(j) (q ∈ domδ ). Fix j with d(x,y)+2−j < r,
q B Σ0 q Y
1
l ∈ N with H(j)(p) ≥ 1 and x ∈ α H(j)(p) −1 and N ∈ N with H(j)(domδ ∩pN.B) ⊆
q l q l q
H(j)(p)l+1.B. Any p′ ∈ domδ∩pN.B(cid:16) satisfies (cid:17)
q
u(j)(p′) = (δ ◦H(j))(p′) ⊇ {α H(j)(p) −1 | l′ ≤ l∧H(j)(p) ≥ 1}
q Σ0 q q l′ q l′
1
where the last set contains the point[x. B(cid:16)y density of ((cid:17)p(i)) ⊆ domδ, pick i with p(i) ∈
i
domδ∩pN.B and d(δ(p(i)),y)+2−j < r. Then
x ∈ u(j)(p(i))= b′(hi,ji,q) (⊆ B(δ(p(i));2−j))
q
and hi,ji ⊏′ s. This completes the proof of Property 1.
Finally we show B : Y ⇒ (Σ0(X)2)N ×(N2)N is computable. Fix h ∈ R(1) such that
1
imh = {ha,bi |a ⊏′ b} and consider as a realiser the map I :⊆ B → B defined by
I(q) := hhr(0),r(1),...i,hs(0),s(1),...ii
where r(hi,ji) = G(p(i),j.0ω,q) and s(k) = hπ h(k).0ω,π h(k).0ωi (i,j,k ∈ N). That is, take
1 2
(Ui,Vi) := (δΣ20 ◦G)(p(π1i),π2i.0ω,q) = B1(i,q) and hai,bii := h(i) for each i, so (Ui)i gives
1
the basis numbering b′(·,q) and (a ,b ) gives the relation ⊏′ independent of q.
i i i
For a fixed q ∈ domδY, observe (Ui,Vi) ∈ M◦(δ(p(π1i)),π2i,δY(q)) implies δ(p(π1i)) ∈
Ui ⊆ B(δ(p(π1i));2−π2i) and δY(q) ⊆ Ui∪˙Vi. Then ((Ui,Vi)i,(ai,bi)i)∈ (B◦δY)(q) trivially.
((iii)) =⇒ ((iv)): This proof derives from [11, §26.II, Thm 1]. Assume we are given
((V ) ,Y) ∈ domS, ((T ,U ) ,(a ,b ) ) ∈ B(Y) and hp(0),p(1),...i ∈ (δω )−1{(V ) }.
i i i i i∈N k k k Σ0(X) i i∈N
1
(i) (i)
For each i enumerate (0 for each j s.t. p = 0;a +1 for any j,k s.t. p = b +1). By
j k j k
definition of B, ⊏′ defined in (a ,b ) is a refined inclusion of (T ) with respect to α, so
k k k i i
for any b ∈ N and x ∈ X there exists k ∈ N such that x ∈ α(b) implies x ∈ T and b = b.
ak k
10 R.KENNY
Since ∅ =6 imα 6∋ ∅, this implies {b | k ∈ N} = domα = N, so output is infinite for each i
k
— say this output is q(i) ∈ B.
(i)
(∅,∅), if q = 0
Now, let (T ,U ) := j (j ∈ N). We have (∀i)V = T .
i,j i,j (i) i j i,j
((Ta,Ua), if qj = a+1
Also S
W∗ := T ∩ U ⊆ T ∩ (X \T ) = T \ T (i,j ∈ N)
i,j i,j k,l i,j k,l i,j k,l
hk,li<hi,ji hk,li<hi,ji hk,li<hi,ji
\ \ [
are pairwise disjoint with δ -information available uniformly in i, j and the inputs. Then
Σ0
1
W := W∗ (⊆ V , i ∈ N) are pairwise disjoint with a δω -name of (W ) available.
i j i,j i Σ0 i i
1
Finally, any x ∈ ( V )\( W ) = ( T )\( W∗ ) has x 6∈ Y by an argument we
S i i i′ i′ i,j i,j i,j i,j
now elaborate. First, denote Z := T and Z∗ := W∗ (k ∈ N). Then one can
S S k πS1k,π2k Sk π1k,π2k
check
Y ∩ Z ⊆ Z∗ (4.1)
k<l k k<l k
inductively. Namely, assume (4.1) for some l ∈ N (this is trivially true for l = 0). Then
S S
Z∗ = Z ∩ U ⊆ Z \ Z and (∀k′)Y ⊆ T ∪U imply
k k k′<k π1k′,π2k′ k k′<k k′ k′ k′
Y ∩Z ⊆ (Y ∩Z∗)∪ (Y \U ) ⊆ Z∗∪ (Y ∩Z )
T k k Sk′<k π1k′,π2k′ k k′<k k′
for all k ∈ N, in particular
S S
Y ∩ Z ⊆ Z∗∪ (Y ∩Z ) = Z∗∪ (Y ∩Z )⊆ Z∗
k≤l k k≤l k k′<k k′ k≤l k k<l k k≤l k
by inductSive assumpStion.(cid:0)So, wSe established Y(cid:1)= YS∩ iVi =SY ∩ i,jTi,j = YS∩ kZk ⊆
Z∗ = W∗ = W ⊆ V , and in particular (W ) is a cover of Y. This proves
k k i,j i,j i i i i S i i S S
computability of S.
S S S S
At least two implications in Proposition 4.1 could be improved to results concern-
ing Weihrauch reducibility ([7]) between the mentioned operations. If e.g. each operation
M(·,·,Y) (Y ∈ Y) is guaranteed to possess realisers of a given represented class, then a
corresponding enriched representation δ can also be defined. For the purposes of the
Y,M
present paper, we do not study these notions further; in particular, we have not separated
the conditions of computability for N, M, B, S. We mainly consider a situation where
Proposition 4.1 is applied to Y = X (in Section 5 and thereafter in Sections 7 and 8).
5. Zero-dimensional spaces
Less broadly than in Section 4, one can ask what constitutes a useful nonuniform definition
of effectively zero-dimensional computable metric space; more generally, this might be ad-
dressed for closed effectively separable subspaces. In this paper we consider the problem
for Y = X only1. We consider computability of the following operations, again based on
(1)-(6) in Section 1:
S˜ = S˜X :Σ0(X)N ⇒ Σ0(X)N, R = RX :⊆ Σ0(X)N → ∆0(X)N,
1 1 1 1
M :⊆ X ×Σ0(X) ⇒ ∆0(X), N :⊆ Π0(X)2 ⇒ ∆0(X)
1 1 1 1
1ThesubspacecasecouldsubsequentlybetreatedfollowingSection6toaneffectivisationofthetheorem
on closed subspaces [9, Thm 7.1.8], but we will not do that here.