Slicing points in a pointfree adjunction for $T_D$ 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 $T_D$ axiom, a low order separation axiom between $T_0$ and $T_1$, 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: $\sigma$-frames, $\kappa$-frames, frames, bounded distributive lattices). We establish an adjunction between the category of $T_D$ 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 $T_D$. We introduce sharp partial frames, those for which all points are slicing and characterize these as well as the $T_D$ spatial and strongly $T_D$ spatial partial frames. We conclude with a comparison of sober and $T_D$ 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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