Local compactness and one-point compactification

Try

The real line is not compact, but every point has a compact interval around it. This local availability of compact sets supports constructions that global compactness cannot cover.

One-point compactification adds a single point called infinity. In the real-line example it joins the two distant ends into one point, producing a space homeomorphic to a circle. The new point is topological bookkeeping for leaving every compact subset, not a numerical value larger than all real numbers.

Predict

Predict which subsets should be excluded from a neighborhood of infinity. Compact subsets are the right choice. For discrete N, the compact subsets are exactly the finite ones, so a neighborhood of infinity contains all but finitely many integers.

Predict why the Hausdorff assumption matters. It ensures compact subsets are closed, making their complements open in the original space and allowing old and new neighborhoods to fit one topology.

Experiment

Compare two examples.

Original space	Compact subsets used near infinity	Compactification
Discrete N	Finite subsets	A convergent sequence with its limit
Usual R	Closed bounded subsets	A circle
Noncompact locally compact Hausdorff X	Compact subsets	X plus one point

The table is restricted to the stated hypotheses. Arbitrary non-Hausdorff or non-locally-compact spaces need separate analysis.

Understand

Here locally compact means every point has a compact neighborhood: a compact set containing an open neighborhood of that point. In a Hausdorff space this supplies a base of relatively compact neighborhoods. To see the shrinking mechanism, start with a compact neighborhood K of x and a target open U. Work inside the compact Hausdorff K, which is regular, to shrink around x within U∩interior K. The resulting relative neighborhood lies in interior K and is ambient-open, with compact closure contained in U.

For noncompact locally compact Hausdorff X, let X* be X with one new point. Declare the old opens open, and use complements of compact subsets of X as neighborhoods of infinity. These satisfy the topology axioms. Finite unions of compact sets control finite intersections of new neighborhoods; arbitrary unions containing infinity have complement a closed subset of one compact complement and are therefore compact. Intersections with old opens remain open because compact sets are closed.

An open cover of X* contains a member about infinity, leaving a compact subset of X uncovered. Finitely many additional members cover that remainder, proving compactness. To separate x from infinity, choose a compact neighborhood K containing an open U around x. The disjoint opens U and X* without K do the job. Old points are separated by Hausdorffness of X.

Finally X is dense in X*: a neighborhood of infinity cannot miss all of X, because that would make its compact complement equal X, contradicting noncompactness.

Apply

For discrete N, prove compact subsets are finite using the singleton open cover. Then show the ordinary enumeration converges to infinity because it eventually leaves every finite subset.

Repair “Infinity is approached only by an increasing real sequence.” In the real-line compactification, both n and −n converge to the same added point, and so does any net eventually outside each compact set. The definition is neighborhood-based and does not choose an order direction.

Continue

A later lesson compares this compactification with Stone–Čech, which has a stronger extension property. Optional depth: verify the rational parametrization of the circle in practice and identify its missing point as x tends to either end of the real line. The compact-to-Hausdorff criterion then establishes the extended homeomorphism.