SLIDE 1 Ultrafilters, Closure operators and the Axiom of Choice
Gonc ¸alo Gutierres – CMUC/Universidade de Coimbra It is well known that, in a topological space, the open sets can be characterized using filter
- convergence. In ZF, we cannot replace filters by ultrafilters. It can be proven that the
ultrafilter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ultrafilter Theorem is equivalent to the fact that uX = kX for every topological space X, where k is the usual Kuratowski closure operator and u is the ultrafilter closure, with uX(A) := {x ∈ X : (∃U ultrafilter in X)[U converges to x and A ∈ U]}. These facts arise two different questions that we will try to answer in this talk.
- 1. Under which set theoretic conditions the equality u = k is true in some subclasses of
topological spaces, such as first countable spaces, metric spaces or {R}.
- 2. Is there any topological space X for which uX = kX, but the open sets are
characterized by the ultrafilter convergence?
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ZF – Zermelo-Fraenkel set theory without the Axiom of Choice. ZFC – Zermelo-Fraenkel set theory with the Axiom of Choice.
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ZF – Zermelo-Fraenkel set theory without the Axiom of Choice. ZFC – Zermelo-Fraenkel set theory with the Axiom of Choice. UFT – Ultrafilter Theorem: every filter over a set can extended to an ultrafilter. CUF – Countable Ultrafilter Theorem:
the Ultrafilter Theorem holds for filters with a countable base.
CUF(R) – the Ultrafilter Theorem holds for filters in R with a
countable base.
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ZF – Zermelo-Fraenkel set theory without the Axiom of Choice. ZFC – Zermelo-Fraenkel set theory with the Axiom of Choice. UFT – Ultrafilter Theorem: every filter over a set can extended to an ultrafilter. CUF – Countable Ultrafilter Theorem:
the Ultrafilter Theorem holds for filters with a countable base.
CUF(R) – the Ultrafilter Theorem holds for filters in R with a
countable base.
CC – the Axiom of Countable Choice.
Every countable family of non-empty sets has a choice function.
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Topological spaces
(X, T ) – topological space A ⊆ X Theorem 1 [ZFC] x ∈ A ⇐ ⇒ (∃ U ultrafilter in X)[U → x and A ∈ U]
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Topological spaces
(X, T ) – topological space A ⊆ X Theorem 1 [ZFC] x ∈ A ⇐ ⇒ (∃ U ultrafilter in X)[U → x and A ∈ U] Theorem 2 [ZFC] A ∈ T ⇐ ⇒ [U → x ∈ A = ⇒ A ∈ U]
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Equivalent are:
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Equivalent are: The Ultrafilter Theorem;
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Equivalent are: The Ultrafilter Theorem; For every topological space (X, T ) and A ⊆ X x ∈ A ⇐ ⇒ (∃ U ultrafilter in X)[U → x and A ∈ U] ;
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Equivalent are: The Ultrafilter Theorem; For every topological space (X, T ) and A ⊆ X x ∈ A ⇐ ⇒ (∃ U ultrafilter in X)[U → x and A ∈ U] ; For every topological space (X, T ) and A ⊆ X A ∈ T ⇐ ⇒ [U → x ∈ A = ⇒ A ∈ U] .
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Ultrafilter Closure Operator
uX(A) := {x ∈ X : (∃ U in X)[U → x and A ∈ U]}
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Ultrafilter Closure Operator
uX(A) := {x ∈ X : (∃ U in X)[U → x and A ∈ U]} ˆ uX(A) := {B : A ⊆ B and uX(B) = B}
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Ultrafilter Closure Operator
uX(A) := {x ∈ X : (∃ U in X)[U → x and A ∈ U]} ˆ uX(A) := {B : A ⊆ B and uX(B) = B} kX(A) denotes the usual closure.
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Ultrafilter Closure Operator
uX(A) := {x ∈ X : (∃ U in X)[U → x and A ∈ U]} ˆ uX(A) := {B : A ⊆ B and uX(B) = B} kX(A) denotes the usual closure. Theorem 1 For all X, uX = kX. Theorem 2 For all X, ˆ uX = kX.
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Question
Is the ultrafilter closure idempotent? (u = ˆ u?)
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Question
Is the ultrafilter closure idempotent? (u = ˆ u?) Is there a topological space X for which ˆ uX = kX but uX = kX?
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Question
Is the ultrafilter closure idempotent? (u = ˆ u?) Is there a topological space X for which ˆ uX = kX but uX = kX? The Ultrafilter Theorem is not equivalent to u = ˆ u.
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Diagonal Ultrafilter
UX – the set of all ultrafilters in X. Let X ∈ U 2X and U ∈ UX, X → U if for all A ∈ U, {U ∈ UX : (∃ x ∈ A)[U → x} ∈ X .
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Diagonal Ultrafilter
UX – the set of all ultrafilters in X. Let X ∈ U 2X and U ∈ UX, X → U if for all A ∈ U, {U ∈ UX : (∃ x ∈ A)[U → x} ∈ X . mX(X) := {A ⊆ X : X ∈ U 2A} Proposition [ZF] X → U → x = ⇒ mX(X) → x
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Example
If:
SLIDE 21 Example
If:
- 1. there is a set with no free ultrafilters,
SLIDE 22 Example
If:
- 1. there is a set with no free ultrafilters,
- 2. there is a free ultrafilter on N,
SLIDE 23 Example
If:
- 1. there is a set with no free ultrafilters,
- 2. there is a free ultrafilter on N,
- 3. every infinite set can be mapped onto N;
SLIDE 24 Example
If:
- 1. there is a set with no free ultrafilters,
- 2. there is a free ultrafilter on N,
- 3. every infinite set can be mapped onto N;
then there is a topological space where the ultrafilter closure is not idempotent.
SLIDE 25 Example
If:
- 1. there is a set with no free ultrafilters,
- 2. there is a free ultrafilter on N,
- 3. every infinite set can be mapped onto N;
then there is a topological space where the ultrafilter closure is not idempotent. Is there any model of ZF where these three conditions are satisfied?
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Other classes
The following conditions are equivalente to CUF:
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Other classes
The following conditions are equivalente to CUF: (i) u = k in the class of the first countable spaces;
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Other classes
The following conditions are equivalente to CUF: (i) u = k in the class of the first countable spaces; (ii) ˆ u = k in the class of the first countable spaces;
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Other classes
The following conditions are equivalente to CUF: (i) u = k in the class of the first countable spaces; (ii) ˆ u = k in the class of the first countable spaces; (iii) u = k in the class of the metric spaces;
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Other classes
The following conditions are equivalente to CUF: (i) u = k in the class of the first countable spaces; (ii) ˆ u = k in the class of the first countable spaces; (iii) u = k in the class of the metric spaces; (iv) ˆ u = k in the class of the metric spaces.
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Other classes
The following conditions are equivalente to CUF: (i) u = k in the class of the first countable spaces; (ii) ˆ u = k in the class of the first countable spaces; (iii) u = k in the class of the metric spaces; (iv) ˆ u = k in the class of the metric spaces. CC+ N has a free ultrafilter = ⇒ CUF
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Real space
The following conditions are equivalente to CUF(R):
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Real space
The following conditions are equivalente to CUF(R): (i) uR = kR ;
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Real space
The following conditions are equivalente to CUF(R): (i) uR = kR ; (ii) u = k in the class of the second countable T0-spaces;
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Real space
The following conditions are equivalente to CUF(R): (i) uR = kR ; (ii) u = k in the class of the second countable T0-spaces; (iii) ˆ u = k in the class of the second countable T0-spaces.
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Real space
The following conditions are equivalente to CUF(R): (i) uR = kR ; (ii) u = k in the class of the second countable T0-spaces; (iii) ˆ u = k in the class of the second countable T0-spaces. AC(R)⇒ CC(R)+ N has a free ultrafilter ⇒ CUF(R)
