One-Point Compactification

A topological construction that adjoins a single new point (often written ∞) to a noncompact, locally compact Hausdorff space so that the enlarged space is compact; neighborhoods of the adjoined point are taken to be complements of compact subsets of the original space.

The construction that adjoins a single new point ∞ to a noncompact space X and equips X ∪ {∞} with the topology whose open sets are the opens of X together with sets whose complement in X is compact; for locally compact Hausdorff X this produces a compact Hausdorff space in which X embeds as an open dense subspace.