# A Report on Hausdorff Compactifications of $\mathbb{R}$

## TL;DR

This paper explores the diverse range of Hausdorff compactifications of the real line, demonstrating that there are uncountably many, including well-known and novel examples, highlighting the richness of the compactification landscape.

## Contribution

It provides a comprehensive investigation showing the existence of uncountably many Hausdorff compactifications of r, including explicit examples beyond classical ones.

## Key findings

- There are uncountably many Hausdorff compactifications of r.
- Classical compactifications like Alexandroff, two-point, and Stone-ech are all distinct.
- An explicit example of a new compactification different from known types is constructed.

## Abstract

The goal of this report is to investigate the variety of Hausdorff compactifications of $\mathbb{R}$. The Alexandroff one-point compactification, the two-point compactification, and the Stone-Cech compactification are all clearly different. The ultimate aim is to show that there are in fact uncountably many. An intermediate aim is to exhibit one compactification of $\mathbb{R}$ different from all the compactifications already mentioned.