Transfinite sequences of continuous and Baire 1 functions on separable metric spaces

Baire and weakly Namioka spaces

Recall that a Hausdorff space $X$ is said to be Namioka if for every compact (Hausdorff) space $Y$ and every metric space $Z$, every separately continuous function $f:X\times{Y}\rightarrow{Z}$ is continuous on $D\times{Y}$ for some dense $G_δ$ subset $D$ of $X$. It is well known that in the class of all metrizable spaces, Namioka and Baire spaces coincide (Saint-Raymond, 1983). Further it is known that every completely regular Namioka space is Baire and that every separable Baire space is Namioka (Saint-Raymond, 1983). In our paper we study spaces $X$, we call them weakly Namioka, for which th…

Almost separable spaces

We have defined almost separable space. We show that like separability, almost separability is $c$ productive and converse also true under some restrictions. We establish a Baire Category theorem like result in Hausdorff, Pseudocompacts spaces. We investigate few relationships among separability, almost separability, sequential separability, strongly sequential separability.

Baire property of spaces of $[0,1]$-valued continuous functions

A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. Let $C_p(X,[0,1])$ denote the space of all continuous $[0,1]$-valued functions on a Tychonoff space $X$ with the topology of pointwise convergence. In this paper, we have obtained a characterization when the function space $C_p(X,[0,1])$ is Baire for a Tychonoff space $X$ all separable closed subsets of which are $C$-embedded. In particular, this characterization is true for normal spaces and, hence, for metrizable spaces. Moreover, we obtained that the space $C_p(X,[0,1])$ is Bai…

Codings of separable compact subsets of the first Baire class

Let $X$ be a Polish space and $K$ a separable compact subset of the first Baire class on $X$. For every sequence $\bs$ dense in $\kk$, the descriptive set-theoretic properties of the set \[ \lbf=\{L\in[\nn]: (f_n)_{n\in L} \text{is pointwise convergent}\} \] are analyzed. It is shown that if $K$ is not first countable, then $\lbf$ is $\PB^1_1$-complete. This can also happen even if $K$ is a pre-metric compactum of degree at most two, in the sense of S. Todorcevic. However, if $K$ is of degree exactly two, then $\lbf$ is always Borel. A deep result of G. Debs implies that $\lbf$ contains a Bore…

Baire classification of separately continuous functions and Namioka property

We prove the following two results. 1. If $X$ is a completely regular space such that for every topological space $Y$ each separately continuous function $f:X\times Y\to\mathbb R$ is of the first Baire class, then every Lindelöf subspace of $X$ bijectively continuously maps onto a separable metrizable space. 2. If $X$ is a Baire space, $Y$ is a compact space and $f:X\times Y\to\mathbb R$ is a separately continuous function which is a Baire measurable function, then there exists a dense in $X$ $G_δ$-set $A$ such that $f$ is jointly continuous at every point of $A\times Y$ (this gives a positive…

预紧阿贝尔群的可分商群的三个问题

Three Problems on Separable Quotients of Precompact Abelian Groups

本文构造了两种Baire Pontryagin自反群，解决了预紧阿贝尔群可分商群的三个问题，否定了问题1.25，肯定回答了问题3.12，并实现了问题3.14的所有正则性。

Non-separable h-homogeneous absolute F_{σδ}-spaces and G_{δσ}-spaces

Denote by Q(k) a σ-discrete metric weight-homogeneous space of weight k. We give an internal description of the space Q(k)^ω. We prove that the Baire space B(k) is densely homogeneous with respect to Q(k)^ωif k > ω. Properties of some non-separable h-homogeneous absolute F_{σδ}-sets and G_{δσ}-sets are investigated.

Baire-one函数空间的Baire性质

Baire property of space of Baire-one functions

本文刻画了使Baire-one函数空间$B_1(X)$具有Baire性质的Tychonoff空间$X$，证明$γ$-空间对应的$B_1(X)$为Baire空间，解决了相关公开问题，并得出关于可数稠密齐性的相容性结论。

Baire property of some function spaces

A compact space $X$ is called $π$-monolithic if for any surjective continuous mapping $f:X\rightarrow K$ where $K$ is a metrizable compact space there exists a metrizable compact space $T\subseteq X$ such that $f(T)=K$. A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. Let $C_p(X,Y)$ denote the space of all continuous $Y$- valued functions $C(X,Y)$ on a Tychonoff space $X$ with the topology of pointwise convergence. In this paper we have proved that for a totally disconnected space $X$ the space $C_p(X,\{0,1\})$ is Baire if, and …

Baire 一函数空间的 Lindelöf 型性质

Lindelöf-type properties of spaces of Baire-one functions

本文研究逐点收敛拓扑下 Baire 一函数空间的 Lindelöf 型性质，探讨其正规性、外延性与实紧性，证明遗传 Baire 及 Polish 可分可度量化空间和 Luzin 集上实紧性成立，回答 Teng 的两个问题并提出开放问题。

超限 Hausdorff 维数的 $D$-变体

The $D$-Variant of Transfinite Hausdorff Dimension

本文提出超限 Hausdorff 维数的 $D$-变体 $t_{D}HD$，通过 $D$-维数构建，弥补了传统超限归纳维数的缺陷，并证明了其单调性、双 Lipschitz 不变性等性质，成功实现了对无限 Hausdorff 维数度量空间的分类。

The Baire Category of Subsequences and Permutations which preserve Limit Points

Let $\mathcal{I}$ be a meager ideal on $\mathbf{N}$. We show that if $x$ is a sequence with values in a separable metric space then the set of subsequences [resp. permutations] of $x$ which preserve the set of $\mathcal{I}$-cluster points of $x$ is topologically large if and only if every ordinary limit point of $x$ is also an $\mathcal{I}$-cluster point of $x$. The analogue statement fails for all maximal ideals. This extends the main results in [Topology Appl. \textbf{263} (2019), 221--229]. As an application, if $x$ is a sequence with values in a first countable compact space which is $\mat…

How can we recover Baire class one functions?

Let X and Y be separable metrizable spaces, and f:X-->Y be a function. We want to recover f from its values on a small set via a simple algorithm. We show that this is possible if f is Baire class one, and in fact we get a characterization. This leads us to the study of sets of Baire class one functions and to a characterization of the separability of the dual space of an arbitrary Banach space.

On sequential separability of functional spaces

In this paper, we give necessary and sufficient conditions for the space B_1(X) of first Baire class functions on a Tychonoff space X, with pointwise topology, to be (strongly) sequentially separable.

A weak* separable C(K)* space whose unit ball is not weak* separable

We provide a ZFC example of a compact space K such that C(K)* is w*-separable but its closed unit ball is not w*-separable. All previous examples of such kind had been constructed under CH. We also discuss the measurability of the supremum norm on that C(K) equipped with its weak Baire sigma-algebra.

Completely Baire spaces, Menger spaces, and projective sets

W. Hurewicz proved that analytic Menger sets of reals are $σ$-compact and that co-analytic completely Baire sets of reals are completely metrizable. It is natural to try to generalize these theorems to projective sets. This has previously been accomplished by $V = L$ for projective counterexamples, and the Axiom of Projective Determinacy for positive results. For the first problem, the first author, S. Todorcevic, and S. Tokgöz have produced a finer analysis with much weaker axioms. We produce a similar analysis for the second problem, showing the two problems are essentially equivalent. We al…

可分子集上的一致收敛拓扑

The uniform convergence topology on separable subsets

本文研究实值连续函数空间在可分子集上一致收敛拓扑下的稠密性、闭性、Baire性质及序数空间情形的胞腔度。

Decomposing Borel functions using the Shore-Slaman join theorem

Jayne and Rogers proved that every function from an analytic space into a separable metric space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_σ$ set under it is again $F_σ$. Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers theorem at finite and transfinite levels of the hierarchy of Borel functions: For all countable ordinals $α$ …

Baire函数图的特征与性质

Characterizations and Properties of Graphs of Baire Functions

本文研究了Baire函数的图特征，通过证明Baire-1函数的图是可表示为一系列开集的交集，并讨论了更高Baire类函数的类似问题，以及Baire-2函数图的积累集特性。

Between Polish and completely Baire

All spaces are assumed to be separable and metrizable. Consider the following properties of a space $X$. (1) $X$ is Polish. (2) For every countable crowded $Q\subseteq X$ there exists a crowded $Q'\subseteq Q$ with compact closure. (3) Every closed subspace of $X$ is either scattered or it contains a homeomorphic copy of $2^ω$. (4) Every closed subspace of $X$ is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property re…

弱λ-空间的若干函数应用

Some function applications of weak $λ$-spaces

本文在ZFC下构造了多个拓扑空间实例，解决了关于B₁空间的Choquet性质及贝尔性质的相关问题。

On some properties of the space of upper semicontinuous functions

For a Tychonoff space $X$, we will denote by $USC_{p}(X)$ ($B_1(X)$) a set of all real-valued upper semicontinuous functions (a set of all Baire functions of class 1) defined on $X$ endowed with the pointwise convergence topology. In this paper we describe a class of Tychonoff spaces $X$ for which the space $USC_{p}(X)$ is sequentially separable. Unexpectedly, it turns out that this class coincides with the class of spaces for which a stronger form of the sequential separability for the space $B_1(X)$ holds.

Lebesgue measurability of separately continuous functions and separability

It is studied a connection between the separability and the countable chain condition of spaces with the $L$-property (a topological space $X$ has the $L$-property if for every topological space $Y$, separately continuous function $f:X\times Y\to\mathbb R$ and open set $I\subseteq \mathbb R$ the set $f^{-1}(I)$ is a $F_σ$-set). We show that every completely regular Baire space with the $L$-property and the countable chain condition is separable and construct a nonseparable completely regular space with the $L$-property and the countable chain condition. This gives a negative answer to a questi…

关于广义Namioka空间和乘积空间上函数的联合连续性

On generalized Namioka spaces and joint continuity of functions on product of spaces

本文研究了广义Namioka空间的性质，并探讨了在乘积空间上函数的联合连续性问题，主要贡献在于证明了在某些条件下乘积空间为Namioka空间或广义Namioka空间，并讨论了右拓扑群作用的联合连续性条件。

On questions which are connected with Talagrand problem

We prove the following results. 1. If $X$ is a $α$-favourable space, $Y$ is a regular space, in which every separable closed set is compact, and $f:X\times Y\to\mathbb R$ is a separately continuous everywhere jointly discontinuous function, then there exists a subspace $Y_0\subseteq Y$ which is homeomorphic to $β\mathbb N$. 2. There exist a $α$-favourable space $X$, a dense in $β\mathbb N\setminus\mathbb N$ countably compact space $Y$ and a separately continuous everywhere jointly discontinuous function $f:X\times Y\to\mathbb R$. Besides, it was obtained some conditions equivalent to the fact …

关于纤维到映射的半开性及拟分离映射的半开性

On semi-openness of fiber-onto extensions of minimal semiflows and quasi-separable maps

本文研究了极小流的纤维到扩展及拟分离映射的半开性条件，提出了关于半开映射的结构理论和逆极限技术的应用。

X的$Δ_1$-性质等价于$B_1(X)$的Choquet性质

The $Δ_1$-property of $X$ is equivalent to the Choquet property of $B_1(X)$

该研究刻画了Tychonoff空间X的$Δ_1$-性质与点态收敛拓扑下Baire-one函数空间$B_1(X)$的Choquet性质等价，并在ZFC下构造例子回答相关问题。

On scatteredly continuous maps between topological spaces

A map $f:X\to Y$ between topological spaces is defined to be {\em scatteredly continuous} if for each subspace $A\subset X$ the restriction $f|A$ has a point of continuity. We show that for a function $f:X\to Y$ from a perfectly paracompact hereditarily Baire Preiss-Simon space $X$ into a regular space $Y$ the scattered continuity of $f$ is equivalent to (i) the weak discontinuity (for each subset $A\subset X$ the set $D(f|A)$ of discontinuity points of $f|A$ is nowhere dense in $A$), (ii) the $σ$-continuity ($X$ can be written as a countable union of closed subsets on which $f$ is continuous)…