Recently, the concepts of topoframe and topoboolean have been introduced as a generalization of point-free topology, and the relation between topobooleans and complete I-contact algebras (ICAs) has been studied. In this paper, we first introduce the ICA-topoboolean <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_885_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$B_{\tau (C)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mrow> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation>, in which <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_885_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau (C)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is induced from the complete ICA (B, C), and then characterize compact atomic ICA-topobooleans by their point clusters. As an example of the noncompact case, we determine all clusters of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_885_Article_IEq3.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$\big ( \mathcal {P}(\mathbb {R}), C\big )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi mathvariant="script">P</mi> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>C</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, an ICA on the Boolean algebra of the power set of real numbers <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_885_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>. Finally, we generalize the Smirnov compactification theorem from proximity spaces to atomic ICA-topobooleans.

Compact ICA-topobooleans and the Smirnov compactification theorem

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发表日期：

2025年4月7日

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Ali Akbar Estaji,
Rahimeh Pourkhandani,
Mehdi Vatandoost

摘要

Recently, the concepts of topoframe and topoboolean have been introduced as a generalization of point-free topology, and the relation between topobooleans and complete I-contact algebras (ICAs) has been studied. In this paper, we first introduce the ICA-topoboolean $B_{\tau (C)}$ $B_{τ (C)}$ , in which $\tau (C)$ $τ (C)$ is induced from the complete ICA (B, C), and then characterize compact atomic ICA-topobooleans by their point clusters. As an example of the noncompact case, we determine all clusters of $\big ( \mathcal {P}(\mathbb {R}), C\big )$ $(P (R), C)$ , an ICA on the Boolean algebra of the power set of real numbers $\mathbb {R}$ $R$ . Finally, we generalize the Smirnov compactification theorem from proximity spaces to atomic ICA-topobooleans.