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A Construction in Set‐Theoretic Topology by Means of Elementary Substructures
Author(s) -
Bandlow Ingo
Publication year - 1991
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.19910372607
Subject(s) - citation , combinatorics , set (abstract data type) , mathematical economics , computer science , mathematics , topology (electrical circuits) , library science , programming language
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.