<p>Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(X, d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be an ultrametric space, and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$d_H$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mi>H</mi> </msub> </math></EquationSource> </InlineEquation> be the Hausdorff distance on the set <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\bar{{\textbf {B}}}_X$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mrow> <mi mathvariant="bold">B</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mi>X</mi> </msub> </math></EquationSource> </InlineEquation> of all closed balls in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(X, d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Some interconnections between the properties of the spaces <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(X, d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(\bar{{\textbf {B}}}_X, d_H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mover accent="true"> <mrow> <mi mathvariant="bold">B</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mi>X</mi> </msub> <mo>,</mo> <msub> <mi>d</mi> <mi>H</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are described. It has been established that the space <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(\bar{{\textbf {B}}}_X, d_H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mover accent="true"> <mrow> <mi mathvariant="bold">B</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mi>X</mi> </msub> <mo>,</mo> <msub> <mi>d</mi> <mi>H</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has such properties as discreteness, local finiteness, metrical discreteness, completeness, compactness, and local compactness if and only if the space <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(X, d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has those properties. Necessary and sufficient conditions for the separability of the space <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$(\bar{{\textbf {B}}}_X, d_H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mover accent="true"> <mrow> <mi mathvariant="bold">B</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mi>X</mi> </msub> <mo>,</mo> <msub> <mi>d</mi> <mi>H</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> have also been proved.</p>

Hausdorff distance between ultrametric balls

期刊论文
发表日期：

2025年11月1日

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Oleksiy Dovgoshey

摘要

Let $(X, d)$ $(X, d)$ be an ultrametric space, and let $d_H$ $d_{H}$ be the Hausdorff distance on the set $\bar{{\textbf {B}}}_X$ ${\bar{B}}_{X}$ of all closed balls in $(X, d)$ $(X, d)$ . Some interconnections between the properties of the spaces $(X, d)$ $(X, d)$ and $(\bar{{\textbf {B}}}_X, d_H)$ $({\bar{B}}_{X}, d_{H})$ are described. It has been established that the space $(\bar{{\textbf {B}}}_X, d_H)$ $({\bar{B}}_{X}, d_{H})$ has such properties as discreteness, local finiteness, metrical discreteness, completeness, compactness, and local compactness if and only if the space $(X, d)$ $(X, d)$ has those properties. Necessary and sufficient conditions for the separability of the space $(\bar{{\textbf {B}}}_X, d_H)$ $({\bar{B}}_{X}, d_{H})$ have also been proved.