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On countable tightness type properties of spaces of quasicontinuous functions

Alexander V. Osipov

Abstract

In this paper we get characterizations countable tightness, countable fan-tightness and countable strong fan-tightness of spaces of quasicontinuous functions with the topology of pointwise convergence from a open Whyburn $T_2$-space $X$ into the discrete two-point space $\{0, 1\}$ through properties of $X$ determined by selection principles. These properties (e.g. $S_1(K, K)$, $K_Ω$-Lindelofness, $S_1(K_Ω, K_Ω)$) were defined by M. Scheepers and studied in theory of selection principles in the class of metric spaces. For any cardinal number $κ$, we get a functional characterization of $κ^+$-Lusin space in class of separable metrizable spaces through tightness of compact subsets of a space of quasicontinuous real-valued functions with the topology of pointwise convergence.

On countable tightness type properties of spaces of quasicontinuous functions

Abstract

In this paper we get characterizations countable tightness, countable fan-tightness and countable strong fan-tightness of spaces of quasicontinuous functions with the topology of pointwise convergence from a open Whyburn -space into the discrete two-point space through properties of determined by selection principles. These properties (e.g. , -Lindelofness, ) were defined by M. Scheepers and studied in theory of selection principles in the class of metric spaces. For any cardinal number , we get a functional characterization of -Lusin space in class of separable metrizable spaces through tightness of compact subsets of a space of quasicontinuous real-valued functions with the topology of pointwise convergence.
Paper Structure (8 sections, 16 theorems)

This paper contains 8 sections, 16 theorems.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Countable tightness
  4. Countable strong fan-tightness and countable fan-tightness
  5. Tightness of compact subsets
  6. Examples
  7. Remark
  8. Open questions

Key Result

Lemma 3.1

Every uncountable open Whyburn $\mathcal{K}$-Lindelöf space is a Lusin space.

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • proof
  • ...and 18 more