Discontinuous strongly separately continuous functions of several variables and near coherence of two $P$-filters

Abstract

We consider a notion of near coherence of $n$ $P$-filters and show that the near coherence of any $n$ $P$-filters is equivalent to the near coherence of any two $P$-filters. For any filter $u$ on $\mathbb{N}$ by $\mathbb{N}_u$ we denote the space $\mathbb{N}\cup\{u\}$, in which all points from $\mathbb{N}$ are isolated and sets $A\cup\{u\}$, $A\in u$, are neighborhoods of $u$. In the article, the concept of strongly separately finite sets was introduced. For $X=\mathbb{N}_{u_1}\times\dots\times\mathbb{N}_{u_n}$ we prove that the existence of a strongly separately continuous function $f:X\to\mathbb{R}$ with one-point set $\{(u_1,\dots,u_n)\}$ of discontinuity implies the existence of a strongly separately finite set $E\subseteq X$ such that the characteristic function $\chi|_E$ is discontinuous at $(u_1,\dots,u_n)$. Using this fact we proved that the existence of a strongly separately continuous function $f:X_1\times \dots\times X_n\to\mathbb{R}$ on the product of arbitrary completely regular spaces $X_k$ with an one-point set $\{(x_1,\dots,x_n)\}$ of points of discontinuity, where $x_k$ is non-isolated $G_\delta$-point in $X_k$, is equivalent to near coherence of $P$-filters.