General Structures

Abstract

The problem of constructing extensions and relaxations is connected with obtaining representations with the use of some specific transformations of topological spaces. These transformations are very often compactifications of the space of solutions. But it is possible to construct examples for which the corresponding compactifications are impossible (see, in particular, [35, p. 156]). Therefore other (more general) constructions should be found. These constructions mean a topological improvement of the space of solutions of the initial problem too. One of approaches is connected with different localizations of compactifications and is realized by employing perfect or almost perfect [71] mappings. Moreover, we use some analogue of ‘usual’ compactness. For example, we use the notions of countably compact sets in topological space (TS). Finally, below we consider topological constructions in interrelation with measurable structures. Therefore previously it was important to consider set-theoretic notions and some introductory notions of general topology. In this chapter we recall a series of the well known properties. The most of constructions of this chapter are used in the sequel. But some of them are given for completeness of the presentation. When considering auxiliary constructions, it is useful to discuss some interpretations and connections of them with other notions. For example, when introducing filters and ultrafilters of measurable spaces, it is advisable to give the corresponding Stone representations (the space of Stone representation is considered together with a measurable structure). In Section 2.5 we give a common point of view with respect to constructions of continuity and measurability (the last notion is regarded as main). In the sequel we use it in the basic constructions connected with continuity. But the point of view connected with measurability seems to be highly useful for us.