Nearness Posets

Abstract

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.