SLIDE 1 Characterizing Noetherian spaces as a ∆0
2-analogue to compact spaces1
Matthew de Brecht & Arno Pauly
Kyoto University Université Libre de Bruxelles
TOPOSYM 2016
1This work was supported by JSPS Core-to-Core Program, A. Advanced
Research Networks. The first author was supported by JSPS KAKENHI Grant Number 15K15940. The second author was supported by the ERC inVEST (279499) project.
SLIDE 2 Defining Noetherian spaces
Definition
A topological space X is called Noetherian, iff every strictly ascending chain of open sets is finite.
Theorem (GOUBAULT-LARRECQ)
The following are equivalent for a topological space X:
- 1. X is Noetherian, i.e. every strictly ascending chain of open
sets is finite.
- 2. Every strictly descending chain of closed sets is finite.
- 3. Every open set is compact.
- 4. Every subset is compact.
SLIDE 3 Defining Noetherian spaces
Definition
A topological space X is called Noetherian, iff every strictly ascending chain of open sets is finite.
Theorem (GOUBAULT-LARRECQ)
The following are equivalent for a topological space X:
- 1. X is Noetherian, i.e. every strictly ascending chain of open
sets is finite.
- 2. Every strictly descending chain of closed sets is finite.
- 3. Every open set is compact.
- 4. Every subset is compact.
SLIDE 4
Relevance
Noetherian spaces occur as
◮ spectra of Noetherian rings. ◮ Alexandrov topology of well-quasi orders.
SLIDE 5
Relevance
Noetherian spaces occur as
◮ spectra of Noetherian rings. ◮ Alexandrov topology of well-quasi orders.
SLIDE 6
Relevance
Noetherian spaces occur as
◮ spectra of Noetherian rings. ◮ Alexandrov topology of well-quasi orders.
SLIDE 7
Quasi-Polish spaces
Definition
A countably-based space is quasi-Polish, if its topology is induced by a Smyth-complete quasi-metric.
Proposition (de Brecht)
A locally compact sober countably-based space is quasi-Polish.
SLIDE 8
Quasi-Polish spaces
Definition
A countably-based space is quasi-Polish, if its topology is induced by a Smyth-complete quasi-metric.
Proposition (de Brecht)
A locally compact sober countably-based space is quasi-Polish.
SLIDE 9 When is a Noetherian space quasi-Polish?
Theorem
The following are equivalent for a sober Noetherian space X:
- 1. X is countable.
- 2. X is countably-based.
- 3. X is quasi-Polish.
SLIDE 10
Baire Category Theorem in quasi-Polish spaces
Theorem (HECKMANN; BECHER & GRIGORIEFF)
Let X be quasi-Polish. If X =
i∈N Ai with each Ai being Σ0 2,
then there is some i0 such that Ai0 has non-empty interior.
SLIDE 11 When is a quasi-Polish space Noetherian?
Theorem
The following are equivalent for a quasi-Polish space X:
- 1. X is Noetherian.
- 2. Every ∆0
2-cover of X has a finite subcover.
Corollary
A Noetherian quasi-Polish space is TD iff it is finite.
SLIDE 12 When is a quasi-Polish space Noetherian?
Theorem
The following are equivalent for a quasi-Polish space X:
- 1. X is Noetherian.
- 2. Every ∆0
2-cover of X has a finite subcover.
Corollary
A Noetherian quasi-Polish space is TD iff it is finite.
SLIDE 13
Represented spaces and computability
Definition
A represented space X is a pair (X, δX) where X is a set and δX :⊆ NN → X a surjective partial function.
Definition
F :⊆ NN → NN is a realizer of f : X → Y, iff δY(F(p)) = f(δX(p)) for all p ∈ δ−1
X (dom(f)). Abbreviate: F ⊢ f.
NN
F
− − − − → NN δX δY X
f
− − − − → Y
Definition
f : X → Y is called continuous, iff it has a continuous realizer.
SLIDE 14
Represented spaces and computability
Definition
A represented space X is a pair (X, δX) where X is a set and δX :⊆ NN → X a surjective partial function.
Definition
F :⊆ NN → NN is a realizer of f : X → Y, iff δY(F(p)) = f(δX(p)) for all p ∈ δ−1
X (dom(f)). Abbreviate: F ⊢ f.
NN
F
− − − − → NN δX δY X
f
− − − − → Y
Definition
f : X → Y is called continuous, iff it has a continuous realizer.
SLIDE 15
Represented spaces and computability
Definition
A represented space X is a pair (X, δX) where X is a set and δX :⊆ NN → X a surjective partial function.
Definition
F :⊆ NN → NN is a realizer of f : X → Y, iff δY(F(p)) = f(δX(p)) for all p ∈ δ−1
X (dom(f)). Abbreviate: F ⊢ f.
NN
F
− − − − → NN δX δY X
f
− − − − → Y
Definition
f : X → Y is called continuous, iff it has a continuous realizer.
SLIDE 16
The various classes of spaces
Represented spaces QCB0-spaces ∼ = admissibly represented spaces Quasi-Polish spaces Polish spaces
SLIDE 17
Cartesian closure
Observation
We can form function spaces (to be denoted by C(−, −)) in the category of represented spaces by the UTM-theorem/
Definition
Let S = ({⊤, ⊥}, δS) be defined via δS(p) = ⊥ iff p = 0N.
Definition
The space O(X) of open subsets of X is obtained from C(X, S) via identification.
SLIDE 18
Cartesian closure
Observation
We can form function spaces (to be denoted by C(−, −)) in the category of represented spaces by the UTM-theorem/
Definition
Let S = ({⊤, ⊥}, δS) be defined via δS(p) = ⊥ iff p = 0N.
Definition
The space O(X) of open subsets of X is obtained from C(X, S) via identification.
SLIDE 19
Cartesian closure
Observation
We can form function spaces (to be denoted by C(−, −)) in the category of represented spaces by the UTM-theorem/
Definition
Let S = ({⊤, ⊥}, δS) be defined via δS(p) = ⊥ iff p = 0N.
Definition
The space O(X) of open subsets of X is obtained from C(X, S) via identification.
SLIDE 20 Compactness in synthetic topology
Definition
Call a represented space X compact, if isFull : O(X) → S is continuous.
Theorem
The following are equivalent for a represented space X:
- 1. X is compact.
- 2. For any represented space Y, the map
∀ : O(X × Y) → O(Y) mapping R to {y ∈ Y | ∀x ∈ X (x, y) ∈ R} is continuous.
SLIDE 21 Compactness in synthetic topology
Definition
Call a represented space X compact, if isFull : O(X) → S is continuous.
Theorem
The following are equivalent for a represented space X:
- 1. X is compact.
- 2. For any represented space Y, the map
∀ : O(X × Y) → O(Y) mapping R to {y ∈ Y | ∀x ∈ X (x, y) ∈ R} is continuous.
SLIDE 22
∆0
2-truth values
Definition
Let the represented space S∇ have the points {⊤, ⊥} and the representation ρ(w0ω) = ⊥ and ρ(w1ω) = ⊤.
Definition
We can represent the ∆0
2-subsets of X via their continuous
characteristic functions C(X, S∇).
SLIDE 23
∆0
2-truth values
Definition
Let the represented space S∇ have the points {⊤, ⊥} and the representation ρ(w0ω) = ⊥ and ρ(w1ω) = ⊤.
Definition
We can represent the ∆0
2-subsets of X via their continuous
characteristic functions C(X, S∇).
SLIDE 24 ∇-compactness
Definition
Call a represented space X ∇-compact, if isFull : ∆0
2(X) → S∇
is continuous.
Theorem
The following are equivalent for a represented space X:
- 1. X is ∇-compact.
- 2. For any represented space Y, the map
∀ : ∆0
2(X × Y) → ∆0 2(Y) mapping R to
{y ∈ Y | ∀x ∈ X (x, y) ∈ R} is continuous.
- 3. For any represented space Y, the map
∃ : ∆0
2(X × Y) → ∆0 2(Y) mapping R to
{y ∈ Y | ∃x ∈ X (x, y) ∈ R} is continuous.
SLIDE 25 ∇-compactness
Definition
Call a represented space X ∇-compact, if isFull : ∆0
2(X) → S∇
is continuous.
Theorem
The following are equivalent for a represented space X:
- 1. X is ∇-compact.
- 2. For any represented space Y, the map
∀ : ∆0
2(X × Y) → ∆0 2(Y) mapping R to
{y ∈ Y | ∀x ∈ X (x, y) ∈ R} is continuous.
- 3. For any represented space Y, the map
∃ : ∆0
2(X × Y) → ∆0 2(Y) mapping R to
{y ∈ Y | ∃x ∈ X (x, y) ∈ R} is continuous.
SLIDE 26
The main result
Theorem
A quasi-Polish space is Noetherian iff it is ∇-compact.
Definition
Let C(X) denote the space of constructible subsets of X.
Lemma
Let X be a Noetherian Quasi-Polish space. Then id : ∆0
2(X) → C(X)∇ is well-defined and continuous.
SLIDE 27
The main result
Theorem
A quasi-Polish space is Noetherian iff it is ∇-compact.
Definition
Let C(X) denote the space of constructible subsets of X.
Lemma
Let X be a Noetherian Quasi-Polish space. Then id : ∆0
2(X) → C(X)∇ is well-defined and continuous.
SLIDE 28
The main result
Theorem
A quasi-Polish space is Noetherian iff it is ∇-compact.
Definition
Let C(X) denote the space of constructible subsets of X.
Lemma
Let X be a Noetherian Quasi-Polish space. Then id : ∆0
2(X) → C(X)∇ is well-defined and continuous.
SLIDE 29 The preprint
- M. de Brecht. & A. Pauly.
Noetherian Quasi-Polish Spaces. arXiv 1607.07291, 2016.
