On some iteration semigroups

Janusz Brzdęk

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 $f_{α} : X \to Y (α < ω ₁)$ of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set $α < ω ₁ : f_{α} (x) \neq f (x)$ is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of $Σ ⁰_{η}$ 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 $β (f) \leq ω^{ξ ₁} \cdot ω^{ξ ₂}$ 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 $x \in X$ , ${x} = ⋂ {g (x, n) : n \in ℕ}$ and $g (x, n + 1) \subseteq g (x, n)$ for each...

Filling boxes densely and disjointly

J. Schröder (2003)

Commentationes Mathematicae Universitatis Carolinae

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We effectively construct in the Hilbert cube $ℍ = {[0, 1]}^{ω}$ two sets $V, W \subset ℍ$ with the following properties: (a) $V \cap W = \emptyset$ , (b) $V \cup W$ 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 $ℍ$ . In fact, $V = ⋃_{ℕ} V_{i}$ , $W = ⋃_{ℕ} W_{i}$ , 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 $(0, 0, 0, ...) \in V_{i j}$ , $(1, 1, 1, ...) \in W_{i j}$ , (d) $V_{i} \cup W_{i}$ , $i \in ℕ$ 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 $i$ .

On the $k$ -Baire property

Alessandro Fedeli (1993)

Commentationes Mathematicae Universitatis Carolinae

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In this note we show the following theorem: “Let $X$ be an almost $k$ -discrete space, where $k$ is a regular cardinal. Then $X$ is $k^{+}$ -Baire iff it is a $k$ -Baire space and every point- $k$ open cover $𝒰$ of $X$ such that $card (𝒰) \leq k$ is locally- $k$ at a dense set of points.” For $k = ℵ_{0}$ we obtain a well-known characterization of Baire spaces. The case $k = ℵ_{1}$ is also discussed.

Epics in the category of $T_{2}$ k-groups need not have dense range

W. F. Lamartin (1976)

Colloquium Mathematicae

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