Hybrid topologies on the real line
Given A ⊆ ℝ , the Hattori space H(A) is the topological space ( ℝ , τA ) where each a ∈ A has a τA -neighborhood base { ( a − ε , a + ε ) : ε > 0 } and each b ∈ ℝ − A has a τA -neighborhood base { [ b , b + ε ) : ε > 0 } . Thus, τA may be viewed as a hybrid of the Euclidean topology and the l...
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| Format: | Article |
| Language: | English |
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Universitat Politècnica de València
2023-04-01
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| Series: | Applied General Topology |
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| Online Access: | https://polipapers.upv.es/index.php/AGT/article/view/18566 |
| _version_ | 1797851677238231040 |
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| author | Tom Richmond |
| author_facet | Tom Richmond |
| author_sort | Tom Richmond |
| collection | DOAJ |
| description | Given A ⊆ ℝ , the Hattori space H(A) is the topological space ( ℝ , τA ) where each a ∈ A has a τA -neighborhood base { ( a − ε , a + ε ) : ε > 0 } and each b ∈ ℝ − A has a τA -neighborhood base { [ b , b + ε ) : ε > 0 } . Thus, τA may be viewed as a hybrid of the Euclidean topology and the lower-limit topology. We investigate properties of Hattori spaces as well as other hybrid topologies on ℝ using various combinations of the discrete, left-ray, lower-limit, upper-limit, and Euclidean topologies. Since each of these topologies is generated by a quasi-metric on ℝ, we investigate hybrid quasi-metrics which generate these hybrid topologies. |
| first_indexed | 2024-04-09T19:21:42Z |
| format | Article |
| id | doaj.art-dea0afca19d44c92a11530313e411d27 |
| institution | Directory Open Access Journal |
| issn | 1576-9402 1989-4147 |
| language | English |
| last_indexed | 2024-04-09T19:21:42Z |
| publishDate | 2023-04-01 |
| publisher | Universitat Politècnica de València |
| record_format | Article |
| series | Applied General Topology |
| spelling | doaj.art-dea0afca19d44c92a11530313e411d272023-04-05T11:41:08ZengUniversitat Politècnica de ValènciaApplied General Topology1576-94021989-41472023-04-0124115716810.4995/agt.2023.1856617756Hybrid topologies on the real lineTom Richmond0https://orcid.org/0000-0003-1883-8146Western Kentucky University, USAGiven A ⊆ ℝ , the Hattori space H(A) is the topological space ( ℝ , τA ) where each a ∈ A has a τA -neighborhood base { ( a − ε , a + ε ) : ε > 0 } and each b ∈ ℝ − A has a τA -neighborhood base { [ b , b + ε ) : ε > 0 } . Thus, τA may be viewed as a hybrid of the Euclidean topology and the lower-limit topology. We investigate properties of Hattori spaces as well as other hybrid topologies on ℝ using various combinations of the discrete, left-ray, lower-limit, upper-limit, and Euclidean topologies. Since each of these topologies is generated by a quasi-metric on ℝ, we investigate hybrid quasi-metrics which generate these hybrid topologies.https://polipapers.upv.es/index.php/AGT/article/view/18566hybrid topologyhattori topologyquasi-metric |
| spellingShingle | Tom Richmond Hybrid topologies on the real line Applied General Topology hybrid topology hattori topology quasi-metric |
| title | Hybrid topologies on the real line |
| title_full | Hybrid topologies on the real line |
| title_fullStr | Hybrid topologies on the real line |
| title_full_unstemmed | Hybrid topologies on the real line |
| title_short | Hybrid topologies on the real line |
| title_sort | hybrid topologies on the real line |
| topic | hybrid topology hattori topology quasi-metric |
| url | https://polipapers.upv.es/index.php/AGT/article/view/18566 |
| work_keys_str_mv | AT tomrichmond hybridtopologiesontherealline |