An Upper Bound for the Weight of the Fine Uniformity

An Upper Bound for the Weight of the Fine Uniformity

If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi mathvariant="script">U</mi><mo>)</mo></mrow>...

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Bibliographic Details
Main Authors:	Johnny Cuadro, Margarita Gary, Adolfo Pimienta
Format:	Article
Language:	English
Published:	MDPI AG 2025-08-01
Series:	Mathematics
Subjects:	uniform spaces uniform weight fine uniformity <i>Z</i>-embedded <i>C</i><sub>1</sub>-embedded
Online Access:	https://www.mdpi.com/2227-7390/13/15/2511
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Summary:	If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi mathvariant="script">U</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a Hausdorff uniform space, we define the <b>uniform weight</b> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi mathvariant="script">U</mi><mo>)</mo></mrow></semantics></math></inline-formula> as the smallest cardinal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>κ</mi></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">U</mi></semantics></math></inline-formula> has a basis of cardinality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>κ</mi></semantics></math></inline-formula>. An important topological cardinal of a Tychonoff space <i>X</i> is the number of cozero sets of <i>X</i>, which we denote as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula>. It is known that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi mathvariant="script">U</mi><mo>)</mo><mo>≤</mo><mi>z</mi><mo>(</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> for every compatible uniformity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">U</mi></semantics></math></inline-formula> of <i>X</i>. We do not know if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> can be replaced by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We concentrate ourselves in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>(</mo><mi>X</mi><mo>,</mo><msub><mi mathvariant="script">U</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">U</mi><mi>n</mi></msub></semantics></math></inline-formula> is the <b>fine uniformity</b> of <i>X</i>, i.e., the one having the family of normal covers as a basis. We establish upper bounds for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>(</mo><mi>X</mi><mo>,</mo><msub><mi mathvariant="script">U</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> using the character and pseudocharacter in extensions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow></semantics></math></inline-formula> or using the cardinal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We also find some generalizations of the equivalence: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mrow><mo>(</mo><mi>X</mi><mo>,</mo><msub><mi mathvariant="script">U</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>=</mo><msub><mo>ℵ</mo><mn>0</mn></msub></mrow></semantics></math></inline-formula> if and only if <i>X</i> is metrizable and the set of non-isolated points of <i>X</i> is compact.
ISSN:	2227-7390