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Displaying similar documents to “On some iteration semigroups”

On the closure of Baire classes under transfinite convergences

Tamás Mátrai (2004)

Fundamenta Mathematicae

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Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family f α : X Y ( α < ω ) of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set α < ω : f α ( x ) f ( x ) is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of Σ η sets which can be interesting in its own right.

Functions of Baire class one

Denny H. Leung, Wee-Kee Tang (2003)

Fundamenta Mathematicae

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Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies β ( f ) ω ξ · ω ξ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1...

Rudin-like sets and hereditary families of compact sets

Étienne Matheron, Miroslav Zelený (2005)

Fundamenta Mathematicae

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We show that a comeager Π₁¹ hereditary family of compact sets must have a dense G δ subfamily which is also hereditary. Using this, we prove an “abstract” result which implies the existence of independent ℳ ₀-sets, the meagerness of ₀-sets with the property of Baire, and generalizations of some classical results of Mycielski. Finally, we also give some natural examples of true F σ δ sets.

Extension of functions with small oscillation

Denny H. Leung, Wee-Kee Tang (2006)

Fundamenta Mathematicae

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A classical theorem of Kuratowski says that every Baire one function on a G δ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski’s theorem: if Y is a subspace of a metric space X and f is a...

A note on k-c-semistratifiable spaces and strong β -spaces

Li-Xia Wang, Liang-Xue Peng (2011)

Mathematica Bohemica

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Recall that a space X is a c-semistratifiable (CSS) space, if the compact sets of X are G δ -sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a T 2 -space X is a k-c-semistratifiable space if and only if X has a g function which satisfies the following conditions: (1) For each x X , { x } = { g ( x , n ) : n } and g ( x , n + 1 ) g ( x , n ) for each...

Insertion of a Contra-Baire- 1 (Baire- . 5 ) Function

Majid Mirmiran (2019)

Communications in Mathematics

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Necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a Baire- . 5 function between two comparable real-valued functions on the topological spaces that F σ -kernel of sets are F σ -sets.

Filling boxes densely and disjointly

J. Schröder (2003)

Commentationes Mathematicae Universitatis Carolinae

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We effectively construct in the Hilbert cube = [ 0 , 1 ] ω two sets V , W with the following properties: (a) V W = , (b) V W is discrete-dense, i.e. dense in [ 0 , 1 ] D ω , where [ 0 , 1 ] D denotes the unit interval equipped with the discrete topology, (c) V , W are open in . In fact, V = V i , W = W i , where V i = 0 2 i - 1 - 1 V i j , W i = 0 2 i - 1 - 1 W i j . V i j , W i j are basic open sets and ( 0 , 0 , 0 , ... ) V i j , ( 1 , 1 , 1 , ... ) W i j , (d) V i W i , i is point symmetric about ( 1 / 2 , 1 / 2 , 1 / 2 , ... ) . Instead of [ 0 , 1 ] we could have taken any T 4 -space or a digital interval, where the resolution (number of points) increases with i .