On first Baire class functions defined on some classes of nonmetrizable spaces

E. S. Sukhacheva

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 53 (2018), pp. 39-46

Résumé

For first Baire class functions given on Polish spaces, Baire's and Lebesgue's criteria are known. We prove analogs of these theorems for topological spaces that are both hereditarily Lindelöf and hereditarily Baire spaces. An analogue of Lebesgue's theorem is as follows: let a space $X$ be a hereditarily Lindelöf space and a function $f: X\to\mathbb{R}$. A function $f$ is a first Baire class function if and only if the inverse image of an open set in $\mathbb{R}$ has type $F_\sigma$. The necessity of the following theorem is true for hereditarily Baire spaces and the proof uses the concept of cliquish functions. We affirm that sufficiency is true for hereditarily Lindelöf spaces. An analogue of Baire's theorem is as follows: let $X$ be a hereditarily Lindelöf and hereditarily Baire space. A function $f: X\to\mathbb{R}$ belongs to the set of first Baire class functions if and only if for any non-empty closed subset $F$ the function $f\mid_F$ has a point of continuity. For a subset $A$ of the real line $\mathbb{R}$, a modification of the Sorgenfrey line $S$ denoted as $S_A$ is defined as follows: neighborhoods of points from $A$ are given by neighborhoods of the right half-open topology, and those in the complement of $A$ are given by neighborhoods of the left half-open topology. For a subset $A$ of the real line $\mathbb{R}$, a Hattori space denoted as $H(A)$ is defined as follows: neighborhoods of points from $A$ are given by usual Euclidean neighborhoods and those in the complement of $A$ are given by neighborhoods of the right half-open topology. In particular, spaces $S=S_\varnothing$, $S_A$, and $H(A)$ satisfy the conditions of the previous two theorems.

Détail

Keywords: Sorgenfrey line, function of the first Baire class, hereditarily Baire space, hereditarily Lindelöf space, cliquish function, $F_\sigma$ and $G_\delta$ sets.

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