§2.7. Locally Compact Spaces A topological space X is said to be locally compact if each x ∈ X has an open neighborhood W with W compact. Example. (a) A compact space is locally compact. (b) Rn is locally compact. (c) Each open set in Rn is locally compact. (d) Each closed set in Rn is locally compact. (e) If U is an open subset of Rn and F is a closed subset of Rn, then U ∩ F is locally compact. Hence { ∈ R2 2 2 ≤ } S := (x1, x2) : x1 + x2 1, x2 > 0 is locally compact. But S ∪ {(1, 0)} is not locally compact. (f) A discrete space is locally compact. (g) Let X be an infinite set with cocountable topology. Then X is not locally compact. Indeed, if x ∈ X and if U is a neighborhood of x, then U = X , which is not compact.

(h) Let Y be a compact Hausdorff space and let y0 ∈ Y .

Then the subspace X = Y \{y0} is locally compact and Hausdorff. We shall show that every locally compact Hausdorff space arises this way. 7.1. Theorem. Let X be a locally compact Hausdorff space and let Y be a set consisting of X and one other point. Then there exists a unique topology for Y such that Y is a compact Hausdorff space and the relative topology for X inherited from Y coincides with the original topology for X . Proof. The point in Y \ X will be labeled ∞, the point at infinity. Consider the family S of those subsets U of Y such that either U is an open subset of X , or ∞ ∈ U and X \ U is a compact subset of X . Then S is a topology for Y , and the relative topology for X inherited from (Y , S ) coincides with the original topology for X .

Let {Uα} be an open cover of Y . Then there is an index α0 ∞ ∈ \ such that Ua0 . Since X Ua0 is compact, there exist a

finite number of sets, Uα1 ,..., UαK , that cover X . Hence ∪k Y = j=0Uαj , and Y is compact. Let x ∈ X . Then there exists an open neighborhood U of x in X such that U is compact. Set V = Y \ U. Then U and V are disjoint open sets in Y with x ∈ U and ∞ ∈ V . It follows that Y is Hausdorff. Let T be another topology for Y that has the properties of the theorem. Since (Y , T ) is Hausdorff, {∞} is T -closed, and hence X ∈ T . This implies that if U is an open subset of X , in the original topology, then U ∈ T . Suppose that ∞ ∈ U and X \ U is compact. Then Y \ U = X \ U is T -closed by Corollary 6.4, and hence U ∈ T . Therefore, S ⊂ T . The identity map f :(Y , T ) → (Y , S ) is continuous and bijective. By Theorem 6.7, f is a homeomorphism. It follows that T = S , so that the topology S is unique. If X is compact, then the space Y is obtained from X by adjoining a point {∞} that is both open and closed. If X is a locally compact Hausdorff space that is not compact, then X is dense in Y : each neighborhood of ∞ contains a point of X . In this case Y is called the one-point compactification of X . A compact space Y is said to be a compactification of X if X is (homeomorphic to) a dense subspace of Y . For instance, the closed unit interval [0, 1] is a compactification of the open interval (0, 1). This compactification is different from the one-point compactification of (0, 1), which is homeomorphic to a circle obtained from [0, 1] by gluing together the two end points. Example. If Q is a bounded open set in Rn, then Q is a compactification of Q. The one-point compactification of Q is obtained by gluing together all boundary points of Q. Consider { ∈ R2 2 2 } the unit disc X = (x1, x2) : x1 + x2 < 1 . The closed { ∈ R2 2 2 ≤ } disc X = (x1, x2) : x1 + x2 1 is a compactification of X . The one-point compactification X is obtained by gluing the boundary points of X into one point. It is intuitively clear that the one-point compactification of X is the unit sphere 2 { ∈ R2 2 2 2 } S = (y1, y2, y3) : y1 + y2 + y3 = 1 . This claim can be proved as follows. Let P = (0, 0, 1) denote the north pole of the sphere. Then S 2 is the one-point compactifiction of S 2 \{P}. We prove that X is homeomorphic to S 2 \{P}. First the function f (x) = x/(1 − |x|) is a homeomorphism from X to R2. Next the stereographic projection Φ: S 2 \{P} → R2 is a homeomorphism. Thus X is homeomorphic to S 2 \ P. The stereographic projection

(x1, x2) = Φ(y1, y2, y3) can be defined geometrically by requiring the three points (1, 0, 0), (y1, y2, y3), and (x1, x2, 0) to be collinear. Thus Φ(y1, y2, y3) = (y1, y2)/(1 − y3).