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>...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-08-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/15/2511 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849770527268274176 |
|---|---|
| author | Johnny Cuadro Margarita Gary Adolfo Pimienta |
| author_facet | Johnny Cuadro Margarita Gary Adolfo Pimienta |
| author_sort | Johnny Cuadro |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-9f0de972b366418ea444fcc2ee36fbc0 |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-9f0de972b366418ea444fcc2ee36fbc02025-08-20T03:02:58ZengMDPI AGMathematics2227-73902025-08-011315251110.3390/math13152511An Upper Bound for the Weight of the Fine UniformityJohnny Cuadro0Margarita Gary1Adolfo Pimienta2Departamento de Ciencias Naturales y Exactas, Universidad de la Costa, Barranquilla 080002, ColombiaPrograma de Matemáticas, Universidad del Atlántico, Barranquilla 080002, ColombiaFacultad de Ciencias Básicas y Biomédicas, Vicerrectoría de Investigación, Universidad Simón Bolívar, Barranquilla 080002, ColombiaIf <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.https://www.mdpi.com/2227-7390/13/15/2511uniform spacesuniform weightfine uniformity<i>Z</i>-embedded<i>C</i><sub>1</sub>-embedded |
| spellingShingle | Johnny Cuadro Margarita Gary Adolfo Pimienta An Upper Bound for the Weight of the Fine Uniformity Mathematics uniform spaces uniform weight fine uniformity <i>Z</i>-embedded <i>C</i><sub>1</sub>-embedded |
| title | An Upper Bound for the Weight of the Fine Uniformity |
| title_full | An Upper Bound for the Weight of the Fine Uniformity |
| title_fullStr | An Upper Bound for the Weight of the Fine Uniformity |
| title_full_unstemmed | An Upper Bound for the Weight of the Fine Uniformity |
| title_short | An Upper Bound for the Weight of the Fine Uniformity |
| title_sort | upper bound for the weight of the fine uniformity |
| topic | uniform spaces uniform weight fine uniformity <i>Z</i>-embedded <i>C</i><sub>1</sub>-embedded |
| url | https://www.mdpi.com/2227-7390/13/15/2511 |
| work_keys_str_mv | AT johnnycuadro anupperboundfortheweightofthefineuniformity AT margaritagary anupperboundfortheweightofthefineuniformity AT adolfopimienta anupperboundfortheweightofthefineuniformity AT johnnycuadro upperboundfortheweightofthefineuniformity AT margaritagary upperboundfortheweightofthefineuniformity AT adolfopimienta upperboundfortheweightofthefineuniformity |