Recent Preprints

1. Henry's distances compared with point-free diameters (with Aleš Pultr and Mark Sioen)
   October 2026.
   Abstract: In his study of Gelfand duality ([3]) S. Henry needed a point-free structure of a metric type. He proposed a distance based on the idea of refining sequences of neighbourhoods of the diagonal (following the original Isbell's definition of metrizability as the existence of a countably based uniformity). This had sufficed for the needs of the original task, but the author asked the natural question what was the relation of this structure to the more geometric diameters used in point-free topology. This is, a.o., analyzed and answered in this paper. The relation is shown to be an adjoint situation. It does not provide a one-to-one correspondence: the distances and diameters are connected by multiplicative estimates. Thus, one sees that the induced uniform and Lipschitz structures coincide, but there is no direct comparison of contractive maps; surprisingly enough, however, there is an agreement even in that. Finally, as an application, we exploit the role of entourages in translating the definition of distance to extend the theory to a non-symmetric variant not offered by the natural diameters.
   
   
2. Basic zero-dimensional spaces: a unifying framework for continuity and openness (with João Areias)
   August 2026.
   Abstract: Following a suggestion in [10], we establish the category of basic zero-dimensional spaces and their basic continuous maps. The objects are closure spaces with a distributive closure system containing a specified meet-base of complemented members that make the category a common generalization of those of closure spaces, zero-dimensional topological spaces, Heyting semilattices, and locales (frames). We treat images, preimages, continuity, openness and closedness of maps between basic zero-dimensional spaces. Among our main results, we have a useful description of preimages for basic continuous maps (analogous to the one for localic preimage maps as coframe homomorphisms), and a Joyal-Tierney type theorem for basic open maps. A novel aspect of the category of basic zero-dimensional spaces is a duality principle that, among other things, enables results for basic closed maps to be obtained for free from those for basic open maps.
   
   
3. Frames and Locales: topology without points, 2nd Edition (with Aleš Pultr)
   March 2026, 575 pages (Springer Basel, in press).
   Table of Contents
   
   
4. Structures: A mathematical reader (with Aleš Pultr)
   September 2025, ~680 pages (submitted for publication).
   Table of Contents
   
   
5. Frobenius identities and geometrical aspects of Joyal-Tierney Theorem (with Aleš Pultr)
   July 2024. pdf file
   Abstract: Open and related maps in the point-free context are studied from a consequently geometric perspective: that is, the opens are concrete well-defined subsets, images of localic maps are set-theoretic images f[U], etc.. We present a short proof of Joyal-Tierney Theorem in this setting, a (geometric) characteristic of localic maps that are just complete, and prove that open localic maps also preserve a natural type of sublocales more general than the open ones. A crucial role is played by Frobenius identities that are briefly discussed also in their general aspects.