A note on the QQQ-topology (original) (raw)

In this note we study Abraham Robinson's Q-topology and consider it as a means of constructing counter examples in topology. We shall be interested almost exclusively with separation and disconnectedness conditions in the Q-topology. For instance, we shall show that the Q-topology for a non-discrete completely regular space is a nondiscrete zero-dimensional space in every enlargement. The reader is assumed to know what is meant by an enlargement in the sense of Robinson, what is meant by an ultraproduct enlargement, and to be familiar with the rudiments of non-standard analysis. A good short introduction is sections 1-6 of [6]. We generalize the Q-topology somewhat by introducing the notion of a *topological space and the Q-topology for a *topological space. This will help in dealing with subspaces and will give a slightly simpler notation. Definition 1: A *topological space in a non-standard model *9W is a pair {X,%), where X is an internal set in *9W and$ ^P(X) is an internal family of sets closed under *finite intersections and internal unions, and which contains 0 and X. If (X,%) is a topological space in a model 3W, then *(X,$) = (*X,*$) is a (standard) *topological space in any enlargement *9W of 9W. If © is an infinite collection of topological spaces in a model 3W, then *© contains a non-standard *topological space for any enlargement *9W of 9W.