MA651 Topology. Lecture 9. Compactness 2.This text is based on the following books:• ”Topology” by James Dugundgji• ”Fundamental concepts of topology” by Peter O’Neil• ”Elements of Mathematics: General Topology” by Nicolas BourbakiI have intentionally made several mistakes in this text. The first homework assignment is to findthem.52 Local compactnessMany of important spaces occurring in analysis are not compact, but instead have a local versionof compactness. For example, Euclidean n−space is not compact, but each point of Enhasa neighborhood whose closure is compact. Calling a subset A of a topological space (X, T )relatively compact whenever its closure A is compact, this local property is formalized in:Definition 52.1. A topological space (X, T ) is locally compact if each point has a relativelycompact neighborhood.Example 52.1. Enis locally compact; Bρn(x, 1) is compact for each x ∈ En. Note that thisexample also shows that a locally compact subset of a Hausdorff space need not be closed.Example 52.2. Any infinite discrete space is locally compact, but not compact.Example 52.3. The set of rationals in E1is not a locally compact space.Example 52.4. Any compact space is locally compact - take X as a neighborhood for every x inDefinition (52.1). This is stated as Theorem (52.1) for reference.Theorem 52.1. If (X, T ) is a compact topological space, then X is locally compact.1Example 52.5. Z+, with the topology of Example (46.4) is not locally compact and not compact. Ifn ∈ Z+, then Z+∩[1, n] is a neighborhood of n which is finite, hence compact, but Z+∩ [1, n] = Z+Equivalent formulations of local compactness are given in the following theorem:Theorem 52.2. The following four properties are equivalent:1. X is a locally compact Hausdorff space.2. For each x ∈ X and each neighborhood U(x), there is a relatively compact open V withx ∈ V ⊂ V ⊂ U.3. For each compact C and open U ⊃ C, there is a relatively compact open V with C ⊂ V ⊂V ⊂ U.4. X has a basis consisting of relatively compact open sets.Proof.(1) ⇒ (2) There is some open W with x ∈ W ⊂ W and W compact. Since W is therefore a regularspace, and W ∩ U is a neighborhood of x in W , there is a set G open in W such thatx ∈ G ⊂ GW⊂ W ∩ U. Now G = E ∩ W , where E is open in X, and the desiredneighborhood of x in X is V = E ∩ W .(2) ⇒ (3) For each c ∈ C find a relatively compact neighborhood V (c) with V (c) ⊂ U; since C iscompact, finitely many of these neighborhoods cover C, and by Proposition (49.1), thisunion has compact closure.(3) ⇒ (4) Let B be the family of all relatively compact open sets in X; since each x ∈ X is compact,(3) asserts that B is a basis.(4) ⇒ (1) is trivial.Please note that Theorem (52.2) is formulated for a Hausdorff space, in the homework please findwhere in the proof we use it, and generalize this theorem for non-Hausdorff spaces.Local compactness is not preserved by continuous surjections. For example, if (X, T ) is anynon-locally compact space, map f : X → X, the identity map. Then f is a (D, T ) continuoussurjection, where D is the discrete topology on X. But, while (X, D ) is locally compact, (X, T )is not by choice.However, a continuous open map onto a Hausdorff space does preserve local compactness:2Theorem 52.3. Let (Y, M ) be a Hausdorff space. Let f : X → Y be a (T , M ) continuous, opensurjection. Let X be T -locally compact, then Y is M -locally compact.Proof. For given y ∈ Y choose x ∈ X so that f(x) = y and choose a relatively compact ne ighbor-hood U(x). Because f is an open map, f(U) is a neighborhood of y, and because f(U) is compact(by Theorem (48.1)), we find f rom f(U) ⊂ f(U) = f(U) that f(U) is compact.In the homework please explain why we need (Y, M ) be a Hausdorff space in Theorem (52.2).Definition 52.2. Let (X, T ) be a topological space and A ⊂ X. Then, A is T -locally compactif and only if A is TA-locally compact.Proposition 52.1. Let (X , T ) be a local compact space and A ⊂ X, then A is locally compactif and only if for any x ∈ A there exists a T -neighborhood V of x such that A ∩ (A ∩ V ) isT -compact.Proof is left as a homework.Example 52.6. [0, 1[ is a locally compact subset of E1. To see this choose x ∈ [0, 1[ and considertwo cases:1. x 6= 0. Then choose δ > 0 such that ]x − δ, x + δ[⊂]0, 1[. Then, ]x − δ, x + δ[ is a T -neighborhood of x, and[0, 1[∩([0, 1[∩]x − δ, x + δ[) = [0, 1[∩([x − δ, x + δ])= [x − δ, x + δ]is compact in E12. x = 0. Use ] −12,12[ for V in Proposition (52.1). We have[0, 1[∩([0, 1[∩] −12,12[) = [0, 1[∩[0,12[ = [0, 1[∩[0,12] = [0,12]and this is compact in E1Example 52.7. A subspace of a locally compact space is not necessarily locally compact. Forexample, the set of irrationals I is not locally compact in E1, although E1is locally compact.To see this, let V be any neighborhood of π. If I ∩ (I ∩ V ) is compact, then I ∩ (I ∩ V ) isbounded and closed. For some ε > 0, ]π−ε, π+ε[⊂ V . Choose any rational y with π−ε < y < π+ε.Then y is a cluster point of I ∩ (I ∩ V ), but y 6∈ I ∩ (I ∩ V ), as y 6∈ I . This means thatI ∩ (I ∩ V ) is not closed after all, a contradiction. Then I ∩ (I ∩ V ) cannot be compact, soI is not locally compact.Of course, there is nothing special about π in this argument - any irrational will do.3As with subspaces, a product of locally compact spaces need not be locally compact. If, however,the coordinate spaces are Hausdorff, and if enough of them are compact, then the product will belocally compact.Theorem 52.4.Y{Yα| α ∈ A } is locally compact if and only if all the Yαare locally compactHausdorff spaces and at most finitely many are not compact.Proof. Assume the condition holds. Given {yα} ∈YαYα, for each of the at most finitely manyindices for which Yαis not compact, choose a relatively c ompact neighborhood Vαi(yαi); thenhVα1, · · · , Vαni is a neighborhood of {yα} and hVα1, · · · , Vαni = hVα1, · · · , Vαni is compact.Conversely, assumeYαYαto be locally compact; since each projection pαis a continuous opensurjection, each Yαis certainly locally compact. But also, choosing any relatively compact openV ⊂YαYα, each pα(V ) is compact, and since pα(V ) = Yαfor all but at most finitely many indicesα, the result follows.There is an important connection between local compactness and Alexandroff’s Theorem. A