The main goal of this section is that of proving a generalized version of the Main Theorem of Computable Analysis. First we will generalize many basic notions from classical computable analysis. After having introduced these tools, we will show how to characterize a class of topological spaces for which a generalized version of the the Main Theorem of Computable Analysis holds.
At this point the reader could be surprised to see that instead of using κ-topologies in generalizing computable analysis we will use standard topological tools. To see why recall that any base of theκ-interval topology overRκ has cardinality 2κ (see Theorem 3.4.23) and that the κ-topology onκκ generated by the
classical base has cardinality κ(this is true for any base of cardinality κ). This means that, there will be no representation of Rκ whose induced topology is the κ-interval topology over Rκ (see Section 2.5). As
we have seen in Section 2.5, the fact that the topology induced by the representation we are using is the one we want to work with is essential for computable analysis. As we will see, using topologies we will be able to define representations whose induced topology is the interval topology over Rκ. Note that, since
theκ-interval topology overRκ is contained in the interval topology, every set of objects we introduced in
Section 3.3 (e.g.,κ-continuous functions) will be representable.
Before we start our generalization let us fix some useful coding functions and some notations. Let (wα)α∈κ
be a sequence of elements ofκ<κ. We will use the following notation:
JwαKα∈κ=w
_
0 w
_
1 . . . .
Now we fix a bijection d·,·e:κ×κ→κand we generalize the tupling functions we have defined in Section 2.5.
Definition 4.2.1 (Tupling). Let a:α→κ be a sequence in κα with α < κ. We will write a
β fora(β)to
reduce the number of parentheses. We define a wrappingfunction ι as follows:
ι(a) = 11J0aβ0Kβ∈α11.
Moreover given w1, w2, . . . inκ<κ andp1, p2. . . ,inκκ, we define:
dw1, p1e=dp1, w1e=ι(w1)p1∈κκ
dw1, . . . , wαe=ι(w1). . . ι(wα)with α∈κ
dp1, . . . , pαe=p1(0). . . pα(0)p1(1). . . pi(1). . . withα∈κ
Moreover, let (pα)α∈κ be a sequence of elements in κκ and(wα)α∈κ be a sequence of elements in κ<κ. We
define
d(wα)α∈κe=Jι(wα)Kα∈κ
and
d(pα)α∈κedα, βe=pα(β)for allα, β∈κ.
Note that the wrapping functionιcan now deal with limit lengths. In particular ifa:α→κis a sequence inκαwithαlimit ordinal, thenι(a) will have lengthα+ 2 and will be the sequence 110a
00a1. . .0aβ0. . .11.
In classical computable analysis, a notation of a set M is a surjective function from ω<ω to M, and a
representation ofM is a surjective function fromωω toM. We can easily generalize these notions toκκ.
Definition 4.2.2 (Notations). Let M be a set and ν:⊆κ<κ →M be a surjective function. We will call ν
a notationof M.
Definition 4.2.3 (Representation). Let M be a set and δM :⊆κκ→M be a surjective function. We will
callδ a representationof M.
Since, as we said, we want to generalize the the Main Theorem of Computable Analysis it is natural to start generalizing the concept of effective topological space and their standard representation.
Definition 4.2.4 (Effective Topological Space). An effective topological space is a triple S = (M, σ, ν)
whereM is a set,σ⊆ P(M)is a family of subsets ofM of cardinality at mostκsuch that
x=y⇔ {A∈σ|x∈A}={A∈σ|y∈A} andν :⊆κ<κ→σ is a notation onσ. We will callτ
S the topology generated by takingσ as a subbase.
Definition 4.2.5 (Standard Representation). Let S= (M, σ, ν)be an effective topological space. We define the standard representationδS :⊆κκ→M ofM as follows:
δS(p) =x⇔ {A∈σ|x∈A}={ν(w)|ι(w)Cp},
wherep∈dom(δS)andι(w)Cpimplies thatw∈dom(ν). We will denote the final topology induced by δS
withτFS.
As we have seen in Section 2.5, for effective topological spaces we have that τS =τFs. This fact turns
out to be crucial in proving the Main Theorem of Computable Analysis. Unfortunately as we will see, this property does not hold in general for effective topological spaces overκκ.
First let us prove thatδS is continuous and open with respect to thefinal topology overS.
Lemma 4.2.6. LetS= (M, σ, ν)be an effective topological space. Then we have: (1) δS is continuous in τFS.
(2) δS is open in τFS.
Proof. Continuity follows directly by the definition. Now letγ∈κ<κ. Note that we can assumeγends with
11, otherwise it is enough to define γ0 =γ_2211 and we would haveδ
S[γ] =δS[γ0]. Moreover if δS[γ] =∅
then it is trivially open, hence we can assumeδS[γ] not empty. We claim that
δS([γ]) =
\
{ν(w)|ι(w)Cγ}.
First we will proveδS([γ])⊆T{ν(w)|ι(w)Cγ}. Takex∈δS([γ]). Then there isp∈[γ] such that
{A∈σ|x∈A}={ν(w)|ι(w)/ p}.
Sinceγ⊂pwe havex∈T{ν(w)|ι(w)
Cγ}.
Now we want to prove thatδS([γ])⊇T{ν(w)|ι(w)Cγ}. Letx∈T{ν(w)|ι(w)Cγ}. Definep=γ_γ0
withγ0=Jι(wi)Ki∈κsuch that
∀i < κ. x∈ν(wi) and∀A∈σ. x∈A⇒ ∃i < κ. ν(wi) =A.
Therefore, we havep∈[γ] andδS(p) =xas desired.
Now we need to show thatT
{ν(w)|ι(w)Cγ} is open inτFS. Let us define the following set:
G={γ0∈κ<κ| ∀ι(w)Cγ∃ι(w0)/ γ0. ν(w) =ν(w0)}.
We claim that
δ−S1(\{ν(w)|ι(w)Cγ}) = [
γ0∈G
[γ0]∩dom(δS).
First we will prove the following:
δ−S1(\{ν(w)|ι(w)Cγ})⊆ [
γ0∈G
[γ0]∩dom(δS).
Let p ∈ δ−S1(T
{ν(w) | ι(w)Cγ}). Then, by definition there exists x in T
SinceδS(p)∈T{ν(w)|ι(w)Cγ}, we have {ν(w)|ι(w)Cγ} ⊆ {A∈σ|x∈A}={ν(w)|ι(w)Cp}, which implies ∀ι(w)/ γ∃ι(w0)/ p. ν(w) =ν(w0). But thenp∈S γ0∈G[γ0]∩dom(δS).
Finally it remains to prove that
δ−S1(\{ν(w)|ι(w)Cγ})⊇ [
γ0∈G
[γ0]∩dom(δS).
Letp∈S
γ0∈G[γ0]∩dom(δS). Thenp∈dom(δS) and there isx∈M such thatδS(p) =x, namely
{A∈σ|x∈A}={ν(w)|ι(w)/ p}.
Now, sincep∈S
γ0∈G[γ0] we have
∀ι(w)/ γ∃ι(w0)/ p. ν(w) =ν(w0).
Hence we have that
{ν(w)|ι(w)Cγ} ⊆ {ν(w)|ι(w)/ p}={A∈σ|x∈A}.
Thereforex∈T{ν(w)|ι(w)
Cγ}as desired. Now note that, since S
γ0∈G[γ0] is open inκκ, we have thatδ−S1(T{ν(w)|ι(w)Cγ}) is open in dom(δS)
as desired.
As we said, in generalized computable analysis the fact thatτS =τFS is not true in general for effective
topological spaces. To see this let us give a better characterization ofτFS.
Lemma 4.2.7. LetS= (M, σ, ν)be an effective topological space. ThenτFS containsτS and is closed under
intersections of less thanκelements ofσ.
Proof. First we want to show thatτFS ⊆τS. Note that it is enough to show thatδS is continuous w.r.t. τS.
For everyX ∈τS we have
δS−1(X) ={p∈dom(δS)|ι(w)Cpfor somewwithν(w)⊆X}.
Therefore,δ−S1(X) is trivially open in dom(δS) andδS is continuous w.r.t. τS.
Now we want to prove thatτFS is closed under intersection of less thanκelements ofσ. LetA⊂σsuch
that|A|< κ. Letγ∈κ<κ be defined as follows
γ=Jι(wa)Ka∈A.
where for alla∈A,ν(wa) =a. Then, as we proved in the previous lemma, we have:
δS([γ]) =
\
{ν(w)|ι(w)Cγ},
and by the fact thatδS is open with respect toτFS, we have that
\
{ν(w)|ι(w)Cγ}
is open inτFS as desired.
It is not hard to see that there are many effective topological spaces for which τS does not have the
necessary closure property to be the final topology induced by their standard representations. Let us illustrate this fact by simple example:
Consider the standard order topology over the ordinal κ. Sinceκ > ω, the interval topologyτS overκis
α < κ, the set {α} is open. Hence, there exists a subset B of intervals with end points in κ such that
{α}=S
(β,β0)∈B(β, β0) and there exists an interval (β, β0)∈B such that (β, β0) ={α}.
Consider the case in whichαis a limit ordinal. By the fact thatα∈(β, β0) we haveβ < α < β0. Since
αis a limit ordinal,β < β+ 1< α < βand β+ 1∈(β, β0).
Now let σ be the set of open intervals in κ, note that σ has cardinality κ. Then there is a notation
ν :⊆κ<κ →σover σ. Consider the effective topological spaceS = (κ, σ, ν). We want to show that τ
FS is
the discrete topology. By the previous theorem we have that the intersections of less thanκopen intervals inκis open in the final topology. Moreover for everyα < κwe have that
{α}= \
β∈α
(β, α+ 1).
Hence for every elementαofκwe have{α} ∈τFS as desired. In conclusion since we proved thatτS is not
the discrete topology, we haveτS 6=τFS.
We will now characterize a subclass of topological spaces for which thisτS =τFS.
Definition 4.2.8(κ-effective Space). Let(M, τ) be a topological space. Then it is κ-effectivew.r.t. σ iffσ
is a subbase ofτ of cardinality at mostκandτ is closed under intersections of strictly less thanκelements of σ.
We will say that (M, τ)is κ-effectiveif there is σsuch that (M, τ)isκ-effective w.r.t. σ.
If (M, τ) isκ-effective w.r.t. σand ν is a notation of σ, we will callS = (M, σ, ν)a κ-effective space. Note that in this case τ=τS.
Now, recall form Section 2.5, that reductions can be used to characterize those representation who are particularly well-behaved. We will now follow this intuition to characterize a class of represented spaces on which we can prove a generalized version of the Main Theorem of Computable Analysis.
We will start generalizing the definition of continuous reduction toκκ:
Definition 4.2.9(Reductions). Letδ:⊆κκ→M andδ0:⊆κκ→M be two representations ofM. Then we
will say thatδcontinuously reducestoδ0, in symbolsδ≤
tδ0 iff there is a continuous functionh:⊆κκ→κκ
such that for everyx∈dom(f),δ(x) =δ0(h(x)).
If δ≤tδ0 andδ0≤tδ we will say thatδ andδ0 are continuously equivalentand we will write δ≡tδ0.
Now, following the classical proof we have:
Lemma 4.2.10. Let M be a set, δ0 :⊆κκ →M andδ1:⊆κκ→M be two representations;oreover, let τ0
and τ1 be respectively the final topology induced by δ0 and δ1. Then δ0 ≤t δ1 implies τ1 ⊆τ0. Moreover,
givenδ00 :⊆κκ→M andδ0
1:⊆κκ→M be other two representations ofM, such thatδ00 ≤tδ0 andδ10 ≤tδ1.
Then every(δ0, δ1)-continuous function is (δ0, δ1)-continuous.
Proof. See the proof of Lemma 2.5.6.
Forκ-effective topologies the standard theory applies. In particular we have thatτS=τFS.
Lemma 4.2.11. Let S= (M, σ, ν)be an effective topological space. Then we have: (1) if (M, τS)isκ-effective w.r.t. σ, thenτS =τFS.
(2) ξ≤tδS for allτFS-continuous functions ξ:⊆κ
κ→M.
(3) for every topological space (M0, τ0)and functionH :⊆M →M0 such that H◦δ
S isτ0-continuous we
have thatH is(τFS, τ
0)-continuous.
Proof. (1) By Lemma 4.2.6 and Lemma 4.2.7, we haveτS ⊆τFS and δS continuous w.r.t. τFS. Hence, it is
enough to show that δS is open inτS. Letγ∈κ<κ, as before we can assumeγ ends with 11. Therefore we
have that
δS([γ]) =
\
(2) Letξ:⊆κκ→M be continuous inτFS. Then for everyp∈κ
<κ andX ∈σthere is α < κsuch that
ξ([pα])⊆X iffξ(p)∈X. Now let (wi)i∈κ be an enumeration ofσ. Forw∈κ<κ, we define the following
Wadge strategy: θ(w)(α) = ( ι(wα) ifξ([w])⊆ν(wα), 2 otherwise. Definef(p)(α)(β) = (S
α∈κh(pα))(γ)(β) whereγ is the smallest such that:
|(sup
α∈κ
h(pα))(γ)|> β.
Since θ is monotone then f is well defined and continuous (note that the function g : β 7→ γ is trivially continuous). The fact thatf translates ξto δS follows by the definition, indeed, f(p) is the list ofwi ∈σ
such thatξ(p)∈ν(wi). Finally, since every small portion of the output of the function only depends on a
small portion of the input,f is continuous as desired.
(3) Let T∈τ0. Then (H◦δS)−1(T) is open in dom(H◦δS). ThereforeδS−1(H−
1(T)) =V ∩dom(H◦δ
S)
for some open setV ⊆κκ. HenceH−1(T) =δS(V∩dom(H◦δS)) =δS[V∩δ−S1(dom(H))] =δS(V)∩dom(H).
Now since δS is an open map therefore δS(V) is open inτFS and H
−1(T) is open in dom(H). Hence H is
continuous.
Hence everyκ-effective topological spaceS has the property thatτS =τFS.
Since continuously equivalent representations share the same final topology, every representation which is continuously equivalent to a standard representation of a κ-effective topological spaceS = (M, σ, ν) has
τS as final topology. Given this, it is natural to consider the following class of representations:
Definition 4.2.12 (κ-admissible Representation). Let (M, τ)be a topological space. Then a representation
δ :⊆κκ → M is κ-admissible w.r.t. τ iff δ is continuous and every continuous function ϕ:⊆κκ →M is
continuously reducible toδ.
Note that as in the classical case if a representation δ of a topological space (M, τ) is continuously equivalent to a standard representation of a κ-effective topological spaceS = (M, σ, ν) withτS =τ, then δ
isκ-admissible.
Finally we are ready to prove a generalized version of the Main Theorem of Computable Analysis.
Theorem 4.2.13 (Generalized Main Theorem of Computable Analysis). For i∈ {0,1}, let (Mi, τi)be an
effective topological space and δi:⊆κκ →Mi be a set ofκ-admissible representation of Mi w.r.t. τi. Then
for any functionf :⊆M1→M0 we have:
f is continuous ⇔f has a continuous realizer.
Proof. Sinceδi≡δSi for someκ-effective topological space with final topologyτi, we can prove the theorem
onδSi instead of δi. Let f be continuous. Then by Lemma 4.2.6 (1) f◦δS1 is continuous and by Lemma
4.2.11 (2)f◦δ1 ≤tδS0. Namely, there is a continuous function f0 such that for every p∈dom(f◦δ1), we
havef(δ1(p)) =δS0(f0(p)). Then gis a continuous realizer off.
On the other hand, letfbe a function with a continuous realizerf0. By definition, for everyp∈dom(δS1),
f(δS1(p)) =δS0(f0(p)) thereforef◦δ1is continuous. Now by Lemma 4.2.11 (3) we have thatf is continuous
as desired.