Some applications of the theory of Katetov order to ideal convergence

摘要

In this paper we establish connections between the theory of Katetov order on ideals on countable sets with ideal convergence in general topological spaces, which are used to study the following questions posed by X.G. Zhou, L. Liu and S. Lin [12]:(1) Whether every finite union of I-closed subsets is I-closed?(2) Must every I-continuous map preserve I-convergence?(3) Must every map preserving I-convergence preserve J-convergence if J subset of I?Our main results include:If I is K-uniform, then every finite union of I-closed subsets is I-closed in any space;there exists a countable zero-dimensional Hausdorff space with character equal to the continuum in which there are two I-closed sets with non-I-closed union for some tall F-sigma-ideal I, whileit is independent of ZFC that for every Hausdorff space X of character less than the continuum and every tall F-sigma-ideal I, finite unions of I-closed subsets are always I-closed in X.These answers Question (1). We show the answer to Question (2) is negative in ZFC and the answer to Question (3) is also negative if we replace J subset of I with J =(K) I (which is weaker than J subset of I). (C) 2020 Elsevier B.V. All rights reserved.