# Small cardinals and small Efimov spaces

## TL;DR

This paper introduces a new cardinal characteristic called the splitting number of the reals, explores its connection to Efimov's problem, and analyzes its implications for the structure of infinite compact Hausdorff spaces.

## Contribution

It defines the splitting number of the reals and investigates its relationship to Efimov's problem, providing new insights into topological and set-theoretic properties.

## Key findings

- The splitting number of the reals is a new cardinal characteristic.
- Connections established between the splitting number and Efimov's problem.
- Implications for the existence of certain types of compact spaces.

## Abstract

We introduce and analyze a new cardinal characteristic of the continuum, the \emph{splitting number of the reals}, denoted $\mathfrak{s}(\mathbb R)$. This number is connected to Efimov's problem, which asks whether every infinite compact Hausdorff space must contain either a non-trivial convergent sequence, or else a copy of $\beta \mathbb N$.