# Existence of strongly proper dyadic subbases

## TL;DR

This paper proves that every locally compact separable metric space admits a strongly proper dyadic subbase, enabling a domain representation that is robust and independent of enumeration.

## Contribution

It establishes the existence of strongly proper dyadic subbases for all locally compact separable metric spaces, advancing the theory of domain representations.

## Key findings

- Every locally compact separable metric space has a strongly proper dyadic subbase.
- Strongly proper dyadic subbases induce admissible domain representations.
- The result is independent of the enumeration of the subbase.

## Abstract

We consider a topological space with its subbase which induces a coding for each point. Every second-countable Hausdorff space has a subbase that is the union of countably many pairs of disjoint open subsets. A dyadic subbase is such a subbase with a fixed enumeration. If a dyadic subbase is given, then we obtain a domain representation of the given space. The properness and the strong properness of dyadic subbases have been studied, and it is known that every strongly proper dyadic subbase induces an admissible domain representation regardless of its enumeration. We show that every locally compact separable metric space has a strongly proper dyadic subbase.