54Dxx Fairly general properties
This subtopic introduces the core ideas in fairly general properties, 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
54D05 Connected and locally connected spaces
Overview
Connected and locally connected spaces. This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D10 Lower separation axioms
Overview
Lower separation axioms. This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D15 Higher separation axioms
Overview
Higher separation axioms. This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
Overview
Noncompact covering properties (paracompact, Lindelöf, etc.). This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D25 Covering dimension of topological spaces
Overview
Covering dimension of topological spaces. This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D30 Compactness in general topology
Overview
Compactness in general topology. This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D35 Extensions of spaces (compactifications, superextensions, completions, etc.)
Overview
Extensions of spaces (compactifications, superextensions, completions, etc.). This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D40 Remainders in general topology
Overview
Remainders in general topology. This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D45 Local compactness, $\sigma$-compactness in general topology
Overview
Local compactness, $\sigma$-compactness in general topology. This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D50 $k$-spaces
Overview
$k$-spaces. This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D55 Sequential spaces in general topology
Overview
Sequential spaces in general topology. This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D60 Realcompactness and realcompactification
Overview
Realcompactness and realcompactification. This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D65 Separability of topological spaces
Overview
Separability of topological spaces. This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D70 Base properties of topological spaces
Overview
Base properties of topological spaces. This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks
54D80 Special constructions of topological spaces
Overview
Special constructions of topological spaces. This topic covers broad topological properties such as compactness, connectedness, normality, and related structural conditions.
Related Wikipedia Page
Wikipedia: Topological property
Useful Links
Key Ideas
- covering and separation properties
- connectedness and compactness families
- interaction among general axioms
Typical Uses
Used to characterize and distinguish large classes of spaces in general topology.
Applications
- Topological classification
- Functional-analytic prerequisites
- Set-theoretic topology
References
Recommended Textbooks