A Construction in Set‐Theoretic Topology by Means of Elementary Substructures

The concept of elementary substructure can be used as a convenient tool for solving various problems in set-theoretic topology. For a good introduction see A.Dow [ 5 ] ; BANDLOW [L] discussed some rather simple applications. In the present paper we introduce a general construction in topology using elementary substructures and give examples which demonstrate the usefullness of this construction. This construction is inspired by a somewhat abstract question: Suppose ( X , T) is a topological space, where T denotes the family of all open subsets of X, and & is an elementary substructure (of a suitable Xe(8)) with X , T E 4. The question is whether there exists a (generally much smaller) natural modification of the space X (denoted by X ( J y t ) ) with respect to AE? The purpose is to define X(&) to be like X. At first sight this is X, = X n AE with one of the two natural topologies to consider on X, (the subspace topology or the topology generated by the family { V n X : V E T n &}). Our concept is to associate with each (uniform) space X and each suitable elementary substructure & a (uniform) space X(&) and a canonical mapping Q,; from X onto X(&). As we shall see, this construction is closely related to various questions in topology.