Baire Class Research Articles

Functions of the first Baire class

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Linear extensions of Baire-alpha functions

In this paper, we extend the main result on linear extensions of some Baire-one functions, obtained recently by W. Sieg, to a larger class of Baire functions.Let A be a nonempty subset of a Hausdorff space X, and let α be an ordinal number <ω1. By Bα(X) we denote the space of all real functions X→R of Baire-class-α. F(A) is the set of functions A→R with a property F, such that F(A) is a linear subspace of Bα(A). We prove Borsuk–Dugundji-type extension theorems: we give an explicit form of a linear extension operator T:F(A)→F(X), where A is an Fσ and Gδ-subset of a normal space X. Our results apply for F=to be piecewise continuous, and F=to be of Baire-class-α. We show that T restricted to the subspace of bounded functions from F(A) is a positive isometry with the supremum norm. In particular, this solves partially an extension problem set in 2005 by Kalenda and Spurný.The same technique of proof can be used to obtain similar results for functions with values in separable normed spaces.

A Complete Topological Classification of the Space of Baire Functions on Ordinals

Considering the spaces Bp[1, α] of all Baire functions x: [1, α] → ℝ on the ordinal segments [1, α] that are endowed with the topology of pointwise convergence, we give a complete topological classification of these spaces.

The Baire Class of Topological Entropy of Non–Autonomous Dynamical Systems

We prove that the lower topological entropy considered as a function on the space of sequences of continuous self–maps of a metric compact space belongs to the second Baire class and the upper one belongs to the fourth Baire class.

Characterizations and properties of graphs of Baire functions

Abstract Let X be a paracompact topological space and Y be a Banach space. In this paper, we will characterize the Baire-1 functions f : X → Y by their graph: namely, we will show that f is a Baire-1 function if and only if its graph gr(f) is the intersection of a sequence ( G n ) n = 1 ∞ $\begin{array}{} \displaystyle (G_n)_{n=1}^{\infty} \end{array}$ of open sets in X × Y such that for all x ∈ X and n ∈ ℕ the vertical section of Gn is a convex set, whose diameter tends to 0 as n → ∞. Afterwards, we will discuss a similar question concerning functions of higher Baire classes and formulate some generalized results in slightly different settings: for example we require the domain to be a metrized Suslin space, while the codomain is a separable Fréchet space. Finally, we will characterize the accumulation set of graphs of Baire-2 functions between certain spaces.

The Baire Classification of Strongly Separately Continuous Functions on ℓ&lt;sub&gt;∞&lt;/sub&gt;

We prove that for any \(\alpha\in[0,\omega_1)\) there exists a strongly separately continuous function \(f:\ell_\infty\rightarrow [0,1]\) such that \(f\) belongs to Baire class \(\alpha+1\), if \(\alpha\) is finite, and Baire class \(\alpha+2\) and \(f\) does not belong to the Baire class \(\alpha\).

On the Composition of Derivatives

We show that there exists a derivative \(f\colon [0,1]\to[0,1]\) such that the graph of \(f\circ f\) is dense in \([0,1]^2\), so not a \(G_\delta\)-set. In particular, \(f\circ f\) is everywhere discontinuous, so not of Baire class 1, and hence it is not a derivative. %neither of Baire class 1 nor a derivative.

Pointfree pointwise convergence, Baire functions, and epimorphisms in truncated archimedean ℓ-groups

We define pointfree pointwise convergence, and use it to define the Baire functions on a locale. The main result is that the Baire functions on a locale coincide with the continuous functions on its P-locale coreflection. Furthermore, we show that the Baire functions on a locale constitute the epicompletion of the continuous functions in the relevant category.The relevant category is T, the category of truncated archimedean ℓ-groups, hereafter nicknamed truncs. T is closely related to the famous category W of unital archimedean ℓ-groups. The universal objects in T are of the form R0L, the trunc of real-valued locale maps L→R which vanish at the designated point of a pointed locale L.We provide an intuitive definition of pointwise convergence in R0L which extends the classical definition, and show that it has a number of nice properties: all homomorphisms and operations of T are pointwise continuous, and a pointwise dense extension is a trunc epimorphism. Conversely, we show that every epic extension G→H has an epic extension H→K such that G is pointwise dense in K.We show that the rich theory of epimorphisms in W carries over to T with only minor modification. In particular, the epicomplete truncs comprise a full monoreflective subcategory, and are characterized as those objects of the form R0P for a P-locale P. In light of these facts, a reformulation of the last clause of the preceding paragraph is that any trunc is pointwise dense in any epicompletion. And a trunc is epicomplete iff it is pointwise complete, i.e., has no proper extension in which it is pointwise dense.Finally, for a given pointed locale L, we define the functions of Baire class α on L in the classical fashion. A function is Baire class 0 if it lies in R0L, and of Baire class β if it is the pointwise limit of a sequence of functions of Baire class α<β. A Baire function on L is a function of Baire class α for some α. Our results can be summarized as follows.TheoremFor a pointed locale L with P-locale coreflectionP⁎L→L, the Baire functions on L are precisely the continuous functions onP⁎L, i.e., those ofR0P⁎L.TheoremThe embeddingR0L→R0P⁎Lis the functorial epicompletion inT.

Feather topologies

Abstract Topologies of special type in a product of two topological spaces are introduced and considered here, in a general case, then in ℝ n to finish on the plane. In the last case, the graph of any function is closed in the considered topology, so it leads to interesting examples of open sets. Our purpose is to study properties of these topologies and connections between them. We prove also that every continuous real function from the plane equipped with the considered topology is in the first class of Baire.

Dynamics of typical Baire-1 functions on the interval

Abstract Let I = [ 0 , 1 ] {I=[0,1]} , and let b ⁢ B 1 {bB_{1}} be the set of Baire-1 self-maps of I. For f ∈ b ⁢ B 1 {f\in bB_{1}} , let Λ ⁢ ( f ) = ⋃ x ∈ I ω ⁢ ( x , f ) {\Lambda(f)=\bigcup_{x\in I}\omega(x,f)} be the set of ω-limit points of f. We prove the following: (i) There exists a residual subset S of b ⁢ B 1 {bB_{1}} such that for any f ∈ S {f\in S} and x ∈ I {x\in I} the ω-limit set ω ⁢ ( x , f ) {\omega(x,f)} is contained in the set of points at which f is continuous, and ω ⁢ ( x , f ) {\omega(x,f)} is an ∞ {\infty} -adic adding machine. (ii) There exists a residual subset S of b ⁢ B 1 {bB_{1}} such that for any f ∈ S {f\in S} and for any ε &gt; 0 {\varepsilon&gt;0} there exists a natural number M such that f m ⁢ ( I ) ⊂ B ε ⁢ ( Λ ⁢ ( f ) ) {f^{m}(I)\subset B_{\varepsilon}(\Lambda(f))} whenever m &gt; M {m&gt;M} . Moreover, f : Λ ⁢ ( f ) → Λ ⁢ ( f ) {f:\Lambda(f)\rightarrow\Lambda(f)} is a bijection.

On structural properties of porouscontinuous functions

Abstract The notion of porouscontinuous function introduced by Borsík and Holos is studied. Relations between porouscontinuous, continuous and ϱ-upper continuous are investigated. Moreover, we show that porouscontinuous functions may not belong to Baire class one.

Some remarks on Baire’s grand theorem

We provide a game theoretical proof of the fact that if f is a function from a zero-dimensional Polish space to $$ \mathbb N^{\mathbb N}$$ that has a point of continuity when restricted to any non-empty compact subset, then f is of Baire class 1. We use this property of the restrictions to compact sets to give a generalisation of Baire’s grand theorem for functions of any Baire class.

Baire classes of Lyapunov invariants

It is shown that no relations exist (apart from inherent ones) between Baire classes of Lyapunov transformation invariants in the compact- open and uniform topologies on the space of linear differential systems. It is established that if a functional on the space of linear differential systems with the compact-open topology is the repeated limit of a multisequence of continuous functionals, then these can be chosen to be determined by the values of system coefficients on a finite interval of the half-line (one for each functional). It is proved that the Lyapunov exponents cannot be represented as the limit of a sequence of (not necessarily continuous) functionals such that each of these depends only on the restriction of the system to a finite interval of the half-line. Bibliography: 28 titles.

Dynamics of typical Baire-1 functions

Let M be the Cantor space or an n-manifold with B1(M,M) the set of Baire-1 self-maps of M. We prove the following:1.For the typical f∈B1(M,M), the maps x⟼ω(x,f) and x⟼τ(x,f) taking x to its ω-limit set and trajectory, respectively, are continuous at a typical point x∈M.2.If s>0, then for the typical (x,f)∈M×B1(M,M), the Hausdorff s-dimensional measure of ω(x,f) is zero.3.If G1 is a residual subset of M, then there is a residual set of points G2⊂M×B1(M,M) all of which generate as their ω-limit set a particular, unique type of adding machine, and if (x,f)∈G2, then ω(x,f)⊂G1.

Typical property of the topological entropy of continuous mappings of compact sets

We show that the topological entropy viewed as a functional on the space of continuous mappings of a metric compact set into itself with the uniform topology is a function of the second Baire class and is lower semicontinuous at a Baire typical point. In particular, we show that the topological entropy is zero at a Baire typical point of the space of continuous mappings of the Baire space of sequences of zeros and units.

Disjoint Borel functions

For each a∈ωω, we define a Baire class one function fa:ωω→ωω which encodes a in a certain sense. We show that for each Borel g:ωω→ωω, fa∩g=∅ implies a∈Δ11(c) where c is any code for g. We generalize this theorem for g in a larger pointclass Γ. Specifically, when Γ=Δ21, a∈L[c]. Also for all n∈ω, when Γ=Δ3+n1, a∈M1+n(c).

On Some Generalizations of Porosity and Porouscontinuity

Abstract In the present paper, we introduce notions of v-porosity and v-porouscontinuity. We investigate some properties of v-poroucontinuity and its connections with porouscontinuity introduced by Borsík and Holos. Moreover, we show that v-porouscontinuous functions may not belong to Baire class one.

On sequential separability of functional spaces

In this paper, we give necessary and sufficient conditions for the space B1(X) of first Baire class functions on a Tychonoff space X, with pointwise topology, to be (strongly) sequentially separable. Also we claim that there are spaces X such that B1(X) is not sequentially separable space, but Cp(X) is sequentially separable (the Sorgenfrey line, the Niemytzki plane).

Classification of bounded Baire class $\xi $ functions

Kechris and Louveau showed that each real-valued bounded Baire class&nbsp;1 function defined on a compact metric space can be written as an alternating sum of a decreasing countable transfinite sequence of upper semicontinuous functions. Moreover, the len

Characterization of order types of pointwise linearly ordered families of Baire class 1 functions

In the 1970s M. Laczkovich posed the following problem: Let B1(X) denote the set of Baire class 1 functions defined on an uncountable Polish space X equipped with the pointwise ordering.Characterize the order types of the linearly orderedsubsets of B1(X). The main result of the present paper is a complete solution to this problem.We prove that a linear order is isomorphic to a linearly ordered family of Baire class 1 functions iff it is isomorphic to a subset of the following linear order that we call ([0,1]↘0<ω1,<altlex), where [0,1]↘0<ω1 is the set of strictly decreasing transfinite sequences of reals in [0,1] with last element 0, and <altlex, the so called alternating lexicographical ordering, is defined as follows: if (xα)α≤ξ,(xα′)α≤ξ′∈[0,1]↘0<ω1 are distinct, and δ is the minimal ordinal where the two sequences differ then we say that(xα)α≤ξ<altlex(xα′)α≤ξ′⇔(δ is even and xδ<xδ′) or (δ is odd and xδ>xδ′).Using this characterization we easily reprove all the known results and answer all the known open questions of the topic.