Discontinuous strongly separately continuous functions of several variables and near coherence of two P-filters
Keywords:
separately continuous function, strongly separately continuous function, P-filter, inverse problem, one-point discontinuityAbstract
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 N by Nu we denote the space N∪{u}, in which all points from N are isolated and sets A∪{u}, A∈u, are neighborhoods of u. In the article, the concept of strongly separately finite sets was introduced. For X=Nu1×⋯×Nun we prove that the existence of a strongly separately continuous function f:X→R with one-point set {(u1,…,un)} of discontinuity implies the existence of a strongly separately finite set E⊆X such that the characteristic function χ|E is discontinuous at (u1,…,un). Using this fact we proved that the existence of a strongly separately continuous function f:X1×⋯×Xn→R on the product of arbitrary completely regular spaces Xk with an one-point set {(x1,…,xn)} of points of discontinuity, where xk is non-isolated Gδ-point in Xk, is equivalent to near coherence of P-filters.