· 2012
Frames and Locales: Topology without points
- 228citations
- 2012year
Short summary
This work introduces and explores the mathematical framework of frames and locales, a point-free approach to topology that redefines spatial concepts without relying on individual points.
AI-generated from the title and abstract; the full text is not read.
Key points
- Introduces frames and locales as a point-free alternative to classical topology.
- Develops core topological concepts like separation axioms and compactness using this new framework.
- Explores advanced topics including uniformity, nearness, and metric frames.
- Discusses the structure of localic morphisms and categories like Loc and Frm.
- Presents the frame of real numbers and real functions within this abstract setting.
AI-generated from the title and abstract; the full text is not read.
Abstract
Preface.- Introduction.- I. Spaces and lattices of open sets.- II. Frames and locales. Spectra.- III. Sublocales.- IV. Structure of localic morphisms. The categories Loc and Frm.- V. Separation axioms.- VI. More on sublocales.-VII. Compactness and local compactness.- VIII. (Symmetric) uniformity and nearness.- IX. Paracompactness.- X. More about completion.- XI. Metric frames.- XII. Entourages, non-symmetric uniformity.- XIII. Connectedness.- XIV. The frame of reals and real functions.- XV. Localic groups.- Appendix I: Posets.- Appendix II: Categories.- Bibliography.- Index of Notation.- Index.
The authors' abstract, as published at the source., 2012 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics