= Two ends of a real-line compactification
{title2=$Y\setminus h[\mathbb R]=C^+\cup C^-$}
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>.
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