S. D. V. Stephens

notes / Math 55b / chapter 3: Completeness, Separation and Quotients / From my notebook

N15 Compactification

(Munkres §29)

Let be a topological space (Hausdorff). I want a compact topological space (which we also want to be Hausdorff) together with an inclusion map which is an embedding, i.e. (with the subspace topology) is a homeomorphism onto its image. So (subspace), and ( dense): is open and dense in .

Example N15.1. We want to compactify ; this is homeomorphic to the unit sphere (the sphere in ). For , the unit circle minus one point, and adding that point back gives . But also compactifies to , which corresponds to . Compactifications are not unique.

Similarly open disc sphere, since , with the missing point as . To compactify, we find something homeomorphic that is compact (if not already compact). The sketch also shows a closed disc and a closed square.

Three compactifications: the circle S^1 as R plus one point at infinity, the interval [0,1] as R plus the two points minus and plus infinity, and the sphere S^2 as R^2 plus a point at infinity.

One-point compactification

If it exists, it is unique up to homeomorphism. Let , where is a new point. We want: compact, and the subspace topology on is , with open in . So:

  • a subset is open in iff ;
  • a subset containing is open in iff is closed, hence compact.

So with compact.

Definition N15.2. is the topology on a one-point compactification.

Theorem N15.3. is a topology on , is compact, is open, and the subspace topology on is .

Remark. However, ( dense in , open) fails if was already compact: then , so is an isolated point of . Also, even if is Hausdorff, might not be.

Proof. is of type 1 and is of type 2. Unions: a union of type 1 sets is type 1. For unions of type 2 sets, so they are still type 2 ( is closed, hence compact?). For mixed unions, where is closed and is compact. It remains to check finite intersections.

is compact: let be an open cover of . Some , and for some compact . Then is an open cover of , so it has a finite subcover: there are with , and . So this is a finite subcover.

The space X covered by small open sets, with one open set A_{i_0} = Y - K containing the added point infinity; the complement K in X is compact.
∎

Definition N15.4. is locally compact if for all there is a compact such that contains a neighborhood of .

A compact set K containing an open neighbourhood U of a point x.

is locally compact; isn’t. is not compact.

Theorem N15.5. The one-point compactification is Hausdorff iff is locally compact and Hausdorff.

Proof. is Hausdorff we can separate points of by open subsets (in or in ). And ∎

End of midterm stuff.