# On n-Hausdorff homogeneous and n-Urysohn homogeneous spaces

## TL;DR

This paper investigates properties of n-Hausdorff and n-Urysohn homogeneous spaces, providing bounds on their size, examples, and constructions of homogeneous extensions, revealing structural limitations and possibilities.

## Contribution

It establishes upper bounds for the cardinality of n-Hausdorff and n-Urysohn homogeneous spaces, and constructs homogeneous extensions from n-Hausdorff spaces.

## Key findings

- No n-Hausdorff 2-homogeneous space exists for n>2
- Provides upper bounds for space cardinalities
- Constructs homogeneous extensions as unions of n-H-closed spaces

## Abstract

In this paper we study $n$-Hausdorff homogeneous and $n$-Urysohn homogeneous spaces. We give some upper bounds for the cardinality of these kind of spaces and give examples. Additionally we show that for every $n>2$, there is no $n$-Hausdorff 2-homogeneous space. Finally, for any $n$-Hausdorff space we construct an $n$-Hausdorff homogeneous extension which is the union of countably many $n$-H-closed spaces.