Normal space

Abstract: In the structure theory of C*-algebras an important role is played by certain topological spaces X which, though locally compact in a certain sense, do not in general satisfy even the weakest separation axiom. This note is concerned with the construction of a compact Hausdorff topology for the space ((X) of all closed subsets of such a space X. This construction occurs naturally in the theory of C*algebras; but, in view of its purely topological nature, it seemed wise to publish it apart from the algebraic context.' A comparison of our topology with the topology of closed subsets studied by Michael in [2] will be made later in this note. For the theory of nets we refer the reader to [1]. A net {x,} is universal if, for every set A, x, is either v-eventually in A or v-eventually outside A. Every net has a universal subnet. By the limit set of a net { x, } of elements of a topological space X we mean the set of those y in X such that { x, } converges to y; the net { x, } is primitive if the limit set of { x, } is the same as the limit set of each subnet of { x, }, i.e., if every cluster point of the net is also a limit of the net. A universal net is obviously primitive. In a locally compact Hausdorff space X the primitive nets are just those which converge either to some point of X or to the point at infinity. An arbitrary topological space X will be called locally compact if, to every poinit x of X and every neighborhood U of x, there is a compact neighborhood of x contained in U. A compact Hausdorff space is of course locally compact; but a compact non-Hausdorff space need not be locally compact. Let X be any fixed topological space (no separation axioms being assumed), and let e(X) be the family of all closed subsets of X (including the void set A). For each compact subset C of X, and each finite family 5 of nonvoid open subsets of X, let U(C; 5) be the subset of e(X) consisting of all Y such that (i) YnGC=A, and (ii) YnA 5tA for each A in W. A subset W of C(X) is open if it is a union of certain of the U(C; 5). It is easily verified that this notion of openness defines a topology for ((X), which we will call the H-topology.

Abstract: 2. J. Nielson, Untersuchungen zur Topologie der geschlossenen Zweiseitigen Flëchen, Acta Math. 50 (1927), Satz 11, 266. 3. R. Baer, Isotopie von Kurven auf orientierbaren geschlossenen Fldchen una ihr Zusammenhang mit der topologischen Deformation der Flachen, Journ. f. Math. 159 (1928), 101. 4. W. Mangier, Die Klassen von topologischen Abbildungen einer geschlossenen Flache auf Sich, Math. Z. 44 (1938), 541, Satz 1, 2,542. 5. R. Fox, On the complementary domains of a certain pair of inequivalent knots, Nederl. Akad. Wetensch. Proc. Ser. A 55-Indag. Math. 14 (1952), 37-40. 6. L. Neuwirth, The algebraic determination of the topological type of the complement of a knot, Proc. Amer. Math. Soc. 12 (1961), 906.

Abstract: Publisher Summary This chapter discusses the theory of Hausdorff convergence. Classically, the Hausdorff distance between two closed subsets in a fixed metric space has been defined. Gromov defined the Hausdorff distance between two metric spaces by taking the infimum of the Hausdorff distances over all ambient spaces into which the two metric spaces are embedded by isometries. There he proved two fundamental results—the precompactness theorem and the convergence theorem—and applied them to the study of the global structures of Riemannian manifolds. The chapter also discusses generalizations of Hausdorff convergence to the case of open manifolds and/or manifolds on which groups act.