Two ends of a real-line compactification ($Y\setminus h[\mathbb{R}]=C^{+}\cup C^{-}$)

For a Hausdorff compactification $h:\mathbb{R}\to Y$, let $C^{\pm }$ be the intersections of the closures of the positive and negative tails, respectively. Each is nonempty and connected by the closed-filter-base theorem. Their union equals the remainder: an image point has a bounded-interval neighbourhood excluding sufficiently long tails, whereas every neighbourhood of a remainder point meets tails by density and compactness of bounded intervals. If the remainder is finite, each $C^{\pm }$ must be a singleton, so there can be only one or two remainder points. In particular there is no three-point compactification of the real line.