On questions which are connected with Talagrand problem

TL;DR

This paper investigates the properties of separately continuous functions on product spaces related to Talagrand's problem, revealing conditions under which certain discontinuous functions exist and characterizing when specific function spaces are Baire.

Contribution

It establishes new results connecting $eta ext{N}$, $eta ext{N}ackslash ext{N}$, and $ ext{α}$-favourable spaces, and characterizes when the function space $C_p(eta ext{N}ackslash ext{N}, ext{{0,1}})$ is Baire.

Findings

Existence of subspaces homeomorphic to $eta ext{N}$ in certain regular spaces.

Construction of separately continuous functions with specific discontinuity properties.

Conditions for the Baire property of the space of continuous functions on $eta ext{N}ackslash ext{N}$.

Abstract

We prove the following results. 1. If $X$ is a $\alpha$-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 $\beta\mathbb N$. 2. There exist a $\alpha$-favourable space $X$, a dense in $\beta\mathbb N\setminus\mathbb N$ 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 that the space $C_p(\beta\mathbb N\setminus\mathbb N,\{0,1\})$ of all continuous functions $x:\beta\mathbb N\setminus\mathbb N\to\{0,1\}$ with the topology of point-wise convergence is a Baire space.