# Nearness Posets

## TL;DR

This paper extends nearness frames to posets for $T_1$ spaces, generalizing Wallman's duality to broader classes of spaces and providing a simplified, sublocale-free admissibility condition.

## Contribution

It introduces a nearness framework for posets representing $T_1$ spaces, extending Wallman's duality to locally compact and completely metrisable spaces.

## Key findings

- Extended Wallman's duality to locally compact $T_1$ spaces
- Generalized duality to completely metrisable spaces
- Provided a sublocale-free admissibility condition

## 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.