A <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ψ</mi> </math></EquationSource> </InlineEquation>-space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Psi ({\mathscr {A}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a Tychonoff space of the form <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\omega }\cup {\mathscr {A}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo>∪</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> is a countable open discrete subspace, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathscr {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> is an almost disjoint family in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> which becomes an uncountable closed discrete subspace in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Psi ({\mathscr {A}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> topologized by the finest topology which makes <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> open and discrete and each <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$A \in {\mathscr {A}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation> is a sequence in <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> converging to the point <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$A \in \Psi (\mathscr {A})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The modification of a <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\Psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ψ</mi> </math></EquationSource> </InlineEquation>-space replacing the points of <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> with certain “building blocks” is called fat <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\Psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ψ</mi> </math></EquationSource> </InlineEquation>-space. Recently, Bonanzinga and Giacopello introduced a “ machine” that associates with a zero-dimensional space having a “bad” family of open sets another zero-dimensional space having an open cover exhibiting similarly “bad” properties, while preserving some of the “good” characteristics of the original space. The construction of the machine is a specific type of fat <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\Psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ψ</mi> </math></EquationSource> </InlineEquation>-space construction. In this paper, we prove that the machine offers a simple and efficient method to distinguish several well-known covering properties and give a partial negative answer to a question posed in 2000 by Bonanzinga and Matveev.

On a specific type of fat $\Psi $-space construction

期刊论文
发表日期：

2026年6月30日

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Maddalena Bonanzinga,
Davide Giacopello

摘要

A $\Psi $ $Ψ$ -space $\Psi ({\mathscr {A}})$ $Ψ (A)$ is a Tychonoff space of the form ${\omega }\cup {\mathscr {A}}$ $ω \cup A$ , where $\omega $ $ω$ is a countable open discrete subspace, $\mathscr {A}$ $A$ is an almost disjoint family in $\omega $ $ω$ which becomes an uncountable closed discrete subspace in $\Psi ({\mathscr {A}})$ $Ψ (A)$ topologized by the finest topology which makes $\omega $ $ω$ open and discrete and each $A \in {\mathscr {A}}$ $A \in A$ is a sequence in $\omega $ $ω$ converging to the point $A \in \Psi (\mathscr {A})$ $A \in Ψ (A)$ . The modification of a $\Psi $ $Ψ$ -space replacing the points of $\omega $ $ω$ with certain “building blocks” is called fat $\Psi $ $Ψ$ -space. Recently, Bonanzinga and Giacopello introduced a “ machine” that associates with a zero-dimensional space having a “bad” family of open sets another zero-dimensional space having an open cover exhibiting similarly “bad” properties, while preserving some of the “good” characteristics of the original space. The construction of the machine is a specific type of fat $\Psi $ $Ψ$ -space construction. In this paper, we prove that the machine offers a simple and efficient method to distinguish several well-known covering properties and give a partial negative answer to a question posed in 2000 by Bonanzinga and Matveev.

On a specific type of fat \(\Psi \)-space construction

摘要