Absolute F spaces Vojt ech Kova r k Charles University, Prague - - PowerPoint PPT Presentation

Absolute Fσδ spaces

Vojtˇ ech Kovaˇ r´ ık

Charles University, Prague vojta.kovarik@gmail.com the first part is a joint work with Ondˇ rej Kalenda

September 12, 2017, Turin

Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute Fσδ spaces September 12, 2017, Turin 1 / 12

Motivation

All our spaces will be Tychonoff.

Absoluteness of low descriptive classes

Let X be a topological space and cX its compactification. Then

1 X is open in cX ⇐

⇒ X is locally compact,

2 X is Gδ in cX ⇐

⇒ X is ˇ Cech-complete,

3 X is closed in cX ⇐

⇒ X is compact,

4 X is Fσ in cX ⇐

⇒ X is σ-compact.

Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute Fσδ spaces September 12, 2017, Turin 2 / 12

Motivation

All our spaces will be Tychonoff.

Absoluteness of low descriptive classes

Let X be a topological space and cX its compactification. Then

1 X is open in cX ⇐

⇒ X is locally compact ⇐ ⇒ X is open in every compactification,

2 X is Gδ in cX ⇐

⇒ X is ˇ Cech-complete ⇐ ⇒ X is Gδ in every compactification,

3 X is closed in cX ⇐

⇒ X is compact ⇐ ⇒ X is closed in every compactification,

4 X is Fσ in cX ⇐

⇒ X is σ-compact ⇐ ⇒ X is Fσ in every compactification. In other words, every closed space (:= closed in some compactification) is absolutely closed (:= closed in every compactification). Analogously for ‘open’, ‘Gδ’ and ‘Fσ’.

Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute Fσδ spaces September 12, 2017, Turin 2 / 12

‘Problems’ with higher classes

An obvious conjecture: the same holds for all descriptive classes. However...

Example (Talagrand, 1985)

There exists an Fσδ space which is not absolutely Fσδ. My topics of interest:

1 When is an Fσδ space absolutely Fσδ?

(first part of the talk)

2 Complexity vs absolute complexity - which combinations are possible

(in ‘X is of class Γ in cX, but of class Ψ in dX’)? (second part of the talk)

3 ...and more questions

(to which I don’t know the answer to yet).

Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute Fσδ spaces September 12, 2017, Turin 3 / 12

Towards the sufficient condition: complete sequences of covers

Definition

A sequence (Cn)n∈N of covers of X is said to be complete, if for every filter F on X, we have (∀n ∈ N)(F ∩ Cn = ∅) = ⇒ (∃x ∈ X)(∀U ∈ U(x))(∀F ∈ F) : U ∩ F = ∅. This notion is connected to descriptive complexity in the following way:

Theorem (Frol´ ık)

1 X is ˇ

Cech-complete ⇐ ⇒ X has a complete sequence of open covers.

2 X is K-analytic ⇐

⇒ X has a complete sequence of countable covers.

3 X is Fσδ ⇐

⇒ X has a complete sequence of countable closed covers.

Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute Fσδ spaces September 12, 2017, Turin 4 / 12

Sufficient condition for a space to be absolutely Fσδ

Theorem (Frol´ ık)

1 X is ˇ

Cech-complete ⇐ ⇒ X has a complete sequence of open covers.

2 X is K-analytic ⇐

⇒ X has a complete sequence of countable covers.

3 X is Fσδ ⇐

⇒ X has a complete sequence of countable closed covers.

Problematic question number one (Frol´ ık)

Describe those topological spaces which are Fσδ in every compactification.

Theorem 1 (Kalenda, K.)

X is absolutely Fσδ ⇐ X has a compl. seq. of countable disjoint Fσ covers. We can get away with less - but it is not a characterization (yet, anyway).

Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute Fσδ spaces September 12, 2017, Turin 5 / 12

Consequences

Definition

A topological space X is said to be hereditarily Lindel¨

• f if every open cover
• f every subspace Y of X has a countable sub-cover.

(in particular, separable metrizable spaces are hereditarily Lindel¨

• f)

Corollary

A hereditarily Lindel¨

• f space which is Fσδ is absolutely Fσδ.

Proposition (Holick´ y, Spurn´ y)

For hereditarily Lindel¨

• f spaces, (F-Borel) complexity is automatically

absolute.

Corollary

Every separable Banach space is absolutely Fσδ (when equipped with weak topology).

Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute Fσδ spaces September 12, 2017, Turin 6 / 12

Second part: F-Borel classes

We will need the following definition:

Definition (F-Borel sets)

We denote F1(X) := closed subsets of X, F2(X) := Fσ subsets of X, F3(X) := Fσδ subsets of X, Fα(X) :=

• β<α

Fβ(X)

• σ

for 1 < α < ω1 even, Fα(X) :=

• β<α

Fβ(X)

• δ

for 1 < α < ω1 odd.

Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute Fσδ spaces September 12, 2017, Turin 7 / 12

Talagrand’s broom spaces

Talagrand has constructed an Fσδ space X, such that for one of its compactifications cX, X does not belong to any of the classes Fα(cX), α < ω1. Based on his construction, we have obtained the following result:

Talagrand’s examples and their properties

For every two countable ordinals α ≥ β ≥ 3, α odd, there exists a space X α

β , such that

1 the complexity of X α

β is Fβ;

2 the absolute complexity of X α

β is Fα.

Notes: By ‘complexity’ we mean that it belongs to the given class, but not to any lower class. The lower bound on the absolute complexity is Talagrand’s.

Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute Fσδ spaces September 12, 2017, Turin 8 / 12

Canonical Fα-sets containing X

Every ‘nice’ (at least K-analytic) space has a monotone Suslin scheme, that is, a sequence of covers (Cn)n satisfying: Cn = {Cs| s ∈ Nn}; For each sequence s: Cs = {Csˆk| k ∈ N}.

The sets Xn for n = 0, 1, 2, . . .

In every compactification cX, we define: X0 := X

cX

X1 :=

n

• s∈NnCs

cX

X2 :=

n

• s∈Nn
• n
• t∈NkCsˆt

cX

And so on for any α < ω1. ‘Huh, what about α ≥ ω?’

Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute Fσδ spaces September 12, 2017, Turin 9 / 12

Admissible mappings and Xα

Sα := ω-ary tree of height α (on N) (Vojta, dont be lazy and draw this)

Definition

A mapping ϕ : Sα → (finite sequences on N) is admissible if it satisfies:

1 ∀s = (s1, s2, . . . , sn): the length of ϕ(s) is s1 + s2 + · · · + sn; 2 ∀s, t : s extends t =

⇒ ϕ(s) extends ϕ(t).

Definition of Xα

Let cX be a fixed compactification. For every α < ω1, we define Xα := {x ∈ cX| ∃ϕ : Sα → S admissible s.t. ∀s ∈ Sα : x ∈ Cϕ(s)

cX}.

To prove the main result, we show that for Talagrand’s broom spaces, X = Xα holds for a suitable α (and α does not depend on the chosen compactification).

Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute Fσδ spaces September 12, 2017, Turin 10 / 12

Problematic question number two

Question 2

Are the ‘nice Fσδ Banach spaces’ (:= Fσδ in second dual) absolutely Fσδ?

Definition

By c0(Γ) we denote the space of long sequences with limit 0: c0(Γ) := {f ∈ RΓ| ∀ǫ > 0 : |f (γ)| ≥ ǫ only holds for finitely many γ ∈ Γ}. c0(Γ) is an example, in some sense canonical, of a nice Fσδ Banach space - but is it absolutely Fσδ? A complication: method used for Talagrand’s broom spaces only works for ‘simple’ spaces, it cannot be applied here.

Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute Fσδ spaces September 12, 2017, Turin 11 / 12

The End

Thank you for your attention!

Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute Fσδ spaces September 12, 2017, Turin 12 / 12