In a space X, if for every countable open cover <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1270_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="135" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {U} =\{U_n:n \in \textbf{N}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">U</mi> <mo>=</mo> <mo stretchy="false">{</mo> <msub> <mi>U</mi> <mi>n</mi> </msub> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="bold">N</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> of X, we can find a subcover <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1270_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="143" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {V}} = \{U_{m_k}:k \in \textbf{N}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">V</mi> <mo>=</mo> <mo stretchy="false">{</mo> <msub> <mi>U</mi> <msub> <mi>m</mi> <mi>k</mi> </msub> </msub> <mo>:</mo> <mi>k</mi> <mo>∈</mo> <mi mathvariant="bold">N</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1270_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="166" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta (\{m_k: U_{m_k} \in {\mathcal {V}} \})=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>:</mo> <msub> <mi>U</mi> <msub> <mi>m</mi> <mi>k</mi> </msub> </msub> <mo>∈</mo> <mi mathvariant="script">V</mi> <mo stretchy="false">}</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> then the space is called a statistically compact space. Extending the recent works of Sarkar, Bal, and Rakshit on statistically compactness, we investigate statistically compactness of a topological space in the star-operator’s background. The concept of star statistically compactness is contrasted to other topological features. This study explains the attributes of star statistically compactness and its subspaces under diverse circumstances, especially under open continuous surjection.

On star statistically compactness

期刊论文
发表日期：

2025年1月17日

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Prasenjit Bal,
Debjani Rakshit,
Susmita Sarkar

摘要

In a space X, if for every countable open cover $\mathcal {U} =\{U_n:n \in \textbf{N}\}$ $U = {U_{n} : n \in N}$ of X, we can find a subcover ${\mathcal {V}} = \{U_{m_k}:k \in \textbf{N}\}$ $V = {U_{m_{k}} : k \in N}$ such that $\delta (\{m_k: U_{m_k} \in {\mathcal {V}} \})=0$ $δ ({m_{k} : U_{m_{k}} \in V}) = 0$ then the space is called a statistically compact space. Extending the recent works of Sarkar, Bal, and Rakshit on statistically compactness, we investigate statistically compactness of a topological space in the star-operator’s background. The concept of star statistically compactness is contrasted to other topological features. This study explains the attributes of star statistically compactness and its subspaces under diverse circumstances, especially under open continuous surjection.