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§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

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§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 x2.7. Locally Compact Spaces A topological space X is said to be locally compact if each x 2 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 f 2 R2 2 2 ≤ g S := (x1; x2) : x1 + x2 1; x2 > 0 is locally compact. But S [ f(1; 0)g 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 2 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 2 Y . Then the subspace X = Y n fy0g 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 n X will be labeled 1, 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 1 2 U and X n 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 fUαg be an open cover of Y . Then there is an index α0 1 2 n 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 2 X . Then there exists an open neighborhood U of x in X such that U is compact. Set V = Y n U. Then U and V are disjoint open sets in Y with x 2 U and 1 2 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, f1g is T -closed, and hence X 2 T . This implies that if U is an open subset of X , in the original topology, then U 2 T . Suppose that 1 2 U and X n U is compact. Then Y n U = X n U is T -closed by Corollary 6.4, and hence U 2 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 f1g 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 1 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 f 2 R2 2 2 g the unit disc X = (x1; x2) : x1 + x2 < 1 . The closed f 2 R2 2 2 ≤ g 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 f 2 R2 2 2 2 g 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 n fPg. We prove that X is homeomorphic to S 2 n fPg. First the function f (x) = x=(1 − jxj) is a homeomorphism from X to R2. Next the stereographic projection Φ: S 2 n fPg ! R2 is a homeomorphism. Thus X is homeomorphic to S 2 n 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)..
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