Wallman Duality for Semilattice Subbases

无格与无点：稳定局部紧空间子基的Vickers对偶

Lattice-Free and Point-Free: Vickers Duality for Subbases of Stably Locally Compact Spaces

本文提出用蕴含关系编码稳定局部紧空间的一般子基，并建立范畴对偶，统一了Priestley-Stone对偶及其多种推广。

Nearness Posets

We extend nearness frames to posets representing bases and even subbases of $T_1$ spaces. This allows us to put a classic duality due to Wallman, between compact $T_1$ spaces and abstract simplicial complexes, into a general nearness framework. Within this framework we extend Wallman's duality to locally compact $T_1$ spaces and, with further modifications, to completely metrisable spaces. Moreover, we provide an elementary sublocale-free version of an admissibility condition due to Picado and Pultr and show how it strengthens Wallman's admissibility condition.

The interplay between weak topologies on topological semilattices

We study the interplay between three weak topologies on a topological semilattice $X$: the weak$^\circ$ topology $\mathcal W^\circ_X$ (generated by the base consiting of open subsemilattices of $X$), the weak$^\bullet$ topology $\mathcal W^\bullet_X$ (generated by the subbase consisting of complements to closed subsemilattices), and the $\mathbb I$-weak topology $\mathcal W_X$ (which is the weakest topology in which all continuous homomorphisms $h:X\to [0,1]$ remain continuous). Also we study the interplay between the weak topologies $\mathcal W^\bullet_X$, $\mathcal W^\circ_X$, $\mathcal W_X$…