Absolute F σδ 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 F σδ 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, ⇒ X is ˇ 2 X is G δ in cX ⇐ Cech-complete, 3 X is closed in cX ⇐ ⇒ X is compact, 4 X is F σ in cX ⇐ ⇒ X is σ -compact. Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute F σδ 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, ⇒ X is ˇ 2 X is G δ in cX ⇐ Cech-complete ⇐ ⇒ X is G δ in every compactification, 3 X is closed in cX ⇐ ⇒ X is compact ⇐ ⇒ X is closed in every compactification, 4 X is F σ in cX ⇐ ⇒ X is σ -compact ⇐ ⇒ X is F σ in every compactification. In other words, every closed space (:= closed in some compactification) is absolutely closed (:= closed in every compactification). Analogously for ‘open’, ‘ G δ ’ and ‘ F σ ’. Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute F σδ 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 F σδ space which is not absolutely F σδ . My topics of interest: 1 When is an F σδ space absolutely F σδ ? (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 F σδ spaces September 12, 2017, Turin 3 / 12
Towards the sufficient condition: complete sequences of covers Definition A sequence ( C n ) n ∈ N of covers of X is said to be complete , if for every filter F on X , we have ( ∀ n ∈ N )( F ∩ C n � = ∅ ) = ⇒ ( ∃ 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 F σδ ⇐ ⇒ X has a complete sequence of countable closed covers. Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute F σδ spaces September 12, 2017, Turin 4 / 12
Sufficient condition for a space to be absolutely F σδ 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 F σδ ⇐ ⇒ X has a complete sequence of countable closed covers. Problematic question number one (Frol´ ık) Describe those topological spaces which are F σδ in every compactification. Theorem 1 (Kalenda, K.) X is absolutely F σδ ⇐ X has a compl. seq. of countable disjoint F σ covers. We can get away with less - but it is not a characterization (yet, anyway). Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute F σδ spaces September 12, 2017, Turin 5 / 12
Consequences Definition A topological space X is said to be hereditarily Lindel¨ of if every open cover of every subspace Y of X has a countable sub-cover. (in particular, separable metrizable spaces are hereditarily Lindel¨ of) Corollary A hereditarily Lindel¨ of space which is F σδ is absolutely F σδ . Proposition (Holick´ y, Spurn´ y) For hereditarily Lindel¨ of spaces, ( F -Borel) complexity is automatically absolute. Corollary Every separable Banach space is absolutely F σδ (when equipped with weak topology). Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute F σδ spaces September 12, 2017, Turin 6 / 12
Second part: F -Borel classes We will need the following definition: Definition ( F -Borel sets) We denote F 1 ( X ) := closed subsets of X , F 2 ( X ) := F σ subsets of X , F 3 ( X ) := F σδ subsets of X , � � F α ( X ) := � F β ( X ) for 1 < α < ω 1 even, β<α σ � � � F α ( X ) := F β ( X ) for 1 < α < ω 1 odd. β<α δ Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute F σδ spaces September 12, 2017, Turin 7 / 12
Talagrand’s broom spaces Talagrand has constructed an F σδ space X , such that for one of its compactifications cX , X does not belong to any of the classes F α ( cX ), α < ω 1 . 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 F β ; 2 the absolute complexity of X α β is F α . 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 F σδ spaces September 12, 2017, Turin 8 / 12
Canonical F α -sets containing X Every ‘nice’ (at least K -analytic) space has a monotone Suslin scheme, that is, a sequence of covers ( C n ) n satisfying: C n = { C s | s ∈ N n } ; For each sequence s : C s = � { C s ˆ k | k ∈ N } . The sets X n for n = 0 , 1 , 2 , . . . In every compactification cX , we define: cX X 0 := X cX X 1 := � � s ∈ N n C s n cX X 2 := � � � t ∈ N k C s ˆ t � n s ∈ N n n And so on for any α < ω 1 . ‘Huh, what about α ≥ ω ?’ Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute F σδ spaces September 12, 2017, Turin 9 / 12
Admissible mappings and X α S α := ω -ary tree of height α (on N ) (Vojta, dont be lazy and draw this) Definition A mapping ϕ : S α → (finite sequences on N ) is admissible if it satisfies: 1 ∀ s = ( s 1 , s 2 , . . . , s n ): the length of ϕ ( s ) is s 1 + s 2 + · · · + s n ; 2 ∀ s , t : s extends t = ⇒ ϕ ( s ) extends ϕ ( t ). Definition of X α Let cX be a fixed compactification. For every α < ω 1 , we define cX } . X α := { x ∈ cX | ∃ ϕ : S α → S admissible s.t. ∀ s ∈ S α : x ∈ C ϕ ( s ) To prove the main result, we show that for Talagrand’s broom spaces, X = X α holds for a suitable α (and α does not depend on the chosen compactification). Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute F σδ spaces September 12, 2017, Turin 10 / 12
Problematic question number two Question 2 Are the ‘nice F σδ Banach spaces’ (:= F σδ in second dual) absolutely F σδ ? Definition By c 0 (Γ) we denote the space of long sequences with limit 0: c 0 (Γ) := { f ∈ R Γ | ∀ ǫ > 0 : | f ( γ ) | ≥ ǫ only holds for finitely many γ ∈ Γ } . c 0 (Γ) is an example, in some sense canonical, of a nice F σδ Banach space - but is it absolutely F σδ ? 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 F σδ spaces September 12, 2017, Turin 11 / 12
The End Thank you for your attention! Vojtˇ ech Kovaˇ r´ ık (MFF UK) Absolute F σδ spaces September 12, 2017, Turin 12 / 12
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