A Report on Hausdorff Compactifications of \(\mathbb{R}\)

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.