54Exx Topological spaces with richer structures
This subtopic introduces the core ideas in topological spaces with richer structures, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.
Specific topics
54E05 Proximity structures and generalizations
Overview
Proximity structures and generalizations. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks
54E15 Uniform structures and generalizations
Overview
Uniform structures and generalizations. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks
54E17 Nearness spaces
Overview
Nearness spaces. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks
54E18 $p$-spaces, $M$-spaces, $\sigma$-spaces, etc.
Overview
$p$-spaces, $M$-spaces, $\sigma$-spaces, etc.. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks
54E20 Stratifiable spaces, cosmic spaces, etc.
Overview
Stratifiable spaces, cosmic spaces, etc.. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks
54E25 Semimetric spaces
Overview
Semimetric spaces. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks
54E30 Moore spaces
Overview
Moore spaces. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks
54E35 Metric spaces, metrizability
Overview
Metric spaces, metrizability. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks
54E40 Special maps on metric spaces
Overview
Special maps on metric spaces. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks
54E45 Compact (locally compact) metric spaces
Overview
Compact (locally compact) metric spaces. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks
54E50 Complete metric spaces
Overview
Complete metric spaces. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks
54E52 Baire category, Baire spaces
Overview
Baire category, Baire spaces. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks
54E55 Bitopological spaces
Overview
Bitopological spaces. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks
54E70 Probabilistic metric spaces
Overview
Probabilistic metric spaces. This topic studies topological spaces equipped with extra structures, such as uniform, proximity, and convergence frameworks.
Related Wikipedia Page
Wikipedia: Topological space
Useful Links
Key Ideas
- topology with additional compatibility data
- uniformity and convergence structures
- enriched continuity notions
Typical Uses
Used when pure topological axioms are supplemented by structure needed for analysis and geometry.
Applications
- Functional analysis
- Topological groups
- Convergence and completion theory
References
Recommended Textbooks