О функциях первого класса Бэра на некоторых классах неметризуемых пространств

Вестник Томского государственного университета. Математика и механика. 2018. № 53. С. 39–46.

Sukhacheva E.

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 →􀁜 . A function f is a first Baire class function if and only if the
inverse image of an open set in 􀁜 has type Fσ .
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 →􀁜 belongs to the set of first Baire class functions if
and only if for any non-empty closed subset F the function f |F has a point of continuity.
For a subset A of the real line 􀁜 , a modification of the Sorgenfrey line S denoted as SA is
defined as follows: neighborhoods of points from A are given by neighborhoods of the right halfopen
topology, and those in the complement of A are given by neighborhoods of the left halfopen
topology. For a subset A of the real line 􀁜 , 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∅ , SA , and H ( A) satisfy the conditions of the previous two theorems.
Keywords: Sorgenfrey line, function of the first Baire class, hereditarily Baire space, hereditarily
Lindelöf space, cliquish function, Fσ and Gδ sets.

Keywords: топология