Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Certain observations on selection principles related to bornological covers using ideals

D. Chandra, P. Das, S. Das

Abstract

We study selection principles related to bornological covers using the notion of ideals. We consider ideals $\mathcal I$ and $\mathcal J$ on $ω$ and standard ideal orderings $KB, K$. Relations between cardinality of a base of a bornology with certain selection principles related to bornological covers are established using cardinal invariants such as modified pseudointersection number, the unbounding number and slaloms numbers. When $\mathcal I \leq_\square \mathcal J$ for ideals $\mathcal I, \mathcal J$ and $\square\in \{1\text{-}1,KB,K\}$, implications among various selection principles related to bornological covers are established. Under the assumption that ideal $\mathcal I$ has a pseudounion we show equivalences among certain selection principles related to bornological covers. Finally, the $\mathcal I\text{-}\mathfrak B^s$-Hurewicz property of $X$ is investigated. We prove that $\mathcal I\text{-}\mathfrak B^s$-Hurewicz property of $X$ coincides with the $\mathfrak B^s$-Hurewicz property of $X$ if $\mathcal I$ has a pseudounion. Implications or equivalences among selection principles, games and $\mathcal I\text{-}\mathfrak B^s$-Hurewicz property which are obtained from our investigations are described in diagrams.

Certain observations on selection principles related to bornological covers using ideals

Abstract

We study selection principles related to bornological covers using the notion of ideals. We consider ideals and on and standard ideal orderings . Relations between cardinality of a base of a bornology with certain selection principles related to bornological covers are established using cardinal invariants such as modified pseudointersection number, the unbounding number and slaloms numbers. When for ideals and , implications among various selection principles related to bornological covers are established. Under the assumption that ideal has a pseudounion we show equivalences among certain selection principles related to bornological covers. Finally, the -Hurewicz property of is investigated. We prove that -Hurewicz property of coincides with the -Hurewicz property of if has a pseudounion. Implications or equivalences among selection principles, games and -Hurewicz property which are obtained from our investigations are described in diagrams.
Paper Structure (8 sections, 32 theorems, 10 equations, 6 figures)

This paper contains 8 sections, 32 theorems, 10 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some observations on Selection Principles
  4. Basic observations on selection principles
  5. Results related to cardinality
  6. Results involving ordering on ideals
  7. Results on selection principles involving ideals having a pseudounion
  8. The $\mathcal{I}$-${\mathfrak{B}^s}$-Hurewicz Property

Key Result

Lemma 3.1

Let $\mathfrak{B}$ be a bornology on a metric space $X$ with a compact base $\mathfrak{B}_0$. Let $f:X\rightarrow Y$ be a continuous function. If $\{U_n:n\in \omega\}$ is an $\mathcal{I}$-$\gamma_{f(\mathfrak{B})^s}$-cover of $X$, then $\{f^{-1}(U_n):n\in \omega\}$ is an $\mathcal{I}$-$\gamma_{\math

Figures (6)

  • Figure 1: Diagram of the selection principles $\binom{\mathcal{A}} {\mathcal{B}}$
  • Figure 2: Diagram of the selection principles when $\mathcal{I}_1\leq_K \mathcal{I}_2$ and $\mathcal{J}_1\leq_{KB} \mathcal{J}_2$
  • Figure 3: Diagram of the selection principles $[\mathcal{A},\mathcal{B}]_\square$
  • Figure 4: Diagram of the selection principles when $\mathcal{I},\mathcal{J}$ have pseudounion
  • Figure 5: Diagram of splittability with respect to ideals $\mathcal{I},\mathcal{J}$
  • ...and 1 more figures

Theorems & Definitions (53)

  • Lemma 3.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • ...and 43 more