# Two extensions of the Stone Duality to the category of zero-dimensional Hausdorff spaces

## TL;DR

This paper extends the Stone Duality Theorem to zero-dimensional Hausdorff spaces, establishing two new duality theorems that encompass extremally disconnected spaces and compactifications, unifying several classical dualities.

## Contribution

It introduces two novel duality theorems for zero-dimensional Hausdorff spaces, extending Stone Duality and deriving related dualities and classical theorems.

## Key findings

- Proves two duality theorems for ZHaus category.
- Derives Tarski Duality and dualities for extremally disconnected spaces.
- Describes categories dually equivalent to zero-dimensional compactifications.

## Abstract

Extending the Stone Duality Theorem, we prove two duality theorems for the category ZHaus of zero-dimensional Hausdorff spaces and continuous maps. Both of them imply easily the Tarski Duality Theorem, as well as two new duality theorems for the category EDTych of extremally disconnected Tychonoff spaces and continuous maps. Also, we describe two categories which are dually equivalent to the category ZComp of zero-dimensional Hausdorff compactifications of zero-dimensional Hausdorff spaces and obtain as a corollary the Dwinger Theorem about zero-dimensional compactifications of a zero-dimensional Hausdorff space.