# Embeddings into countably compact Hausdorff spaces

## TL;DR

This paper investigates which topological spaces can be embedded into countably compact Hausdorff spaces, highlighting limitations related to separation axioms and providing specific counterexamples.

## Contribution

It characterizes embeddings into countably compact Hausdorff spaces and constructs an example of a space with specific separation properties that cannot embed into Urysohn spaces.

## Key findings

- Regular separable scattered space cannot embed into Urysohn countably compact space
- Such spaces can embed into Hausdorff countably compact spaces
- Separation axioms influence embeddability into countably compact spaces

## Abstract

In this paper we consider the problem of characterization of topological spaces that embed into countably compact Hausdorff spaces. We study the separation axioms of subspaces of countably compact Hausdorff spaces and construct an example of a regular separable scattered topological space which cannot be embedded into an Urysohn countably compact topological space but embeds into a Hausdorff countably compact space.