A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff 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.