Slicing points in a pointfree adjunction for TD partial spaces

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.

10.48308/cgasa.2024.235898.1494

Abstract

The TD axiom, a low order separation axiom between T0 and T1, has been of interest to classical topologists for some time; latterly it has also proved interesting to pointfree topologists. Here we investigate it in the context of partial spaces and partial frames (think: σ-frames, κ-frames, frames, bounded distributive lattices). We establish an adjunction between the category of TD partial spaces with continuous maps and the category of partial frames with D-homomorphisms.    Several standard tools (covered primes, right adjoints, point closures) are not appropriate in our setting; we use linked pairs and slicing points instead. Of particular interest are the slicing points of free frames and congruence frames.We examine the fixed objects of the adjunction; both similarities and differences to the classical situation become clear. In particular, there are compact Hausdorff partial spaces that are not TD. We introduce sharp partial frames, those for which all points are slicing and characterize these as well as the TD spatial and strongly TD spatial partial frames. We conclude with a comparison of sober and TD partial spaces.

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References

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Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 02 November 2024

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