A Hausdorff Topology for the Closed Subsets of a Locally Compact Non-hausdorff Space

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 G(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.1 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 [l]. A net {x„} is universal if, for every set A, x, is either ^-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. Ina 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 point 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 Q(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; fJ) be the subset of Q(X) consisting of all F such that (i) Yi\C=A, and (ii) YC\A ¿¿A for each A in $. A subset *W of Q(X) is open if it is a union of certain of the U(C; i). It is easily verified that this notion of openness defines a topology for Q(X), which we will call the H-topology.