Many Valued Topological Spaces

In this chapter we begin the development of the theory of many valued topological spaces. We will base this theory on the B-valued filter monad T F specified in Subsection 2.4.2. In particular, we will introduce many valued topological spaces as topological space objects in the sense of Definition 3.1.1 where the underlying monad is determined by T F . We will give a detailed analysis of the axiom system of many valued topologies including a study of important special cases given for instance by the submonad of stratified (resp. rigid) B*-valued filters. Further, we show that there exist various sources for many valued topological spaces in probability and measure theory as well as in the theory of frames (cf. [57]) and limit spaces (cf. [64, 23]). Many valued topological spaces arise quite naturally in the theory of almost everywhere defined random variables (cf. Subsection 5.3.2) and in the theory of τ-smooth Borel probability measures (cf. Example 5.5.9). Further, for every complete Heyting algebra H there exist adjoint situations between the category of locales, the category of H-valued topological spaces and the category of limit spaces, (cf. Section 5.2 and Subsection 5.4). Finally, every frame can be identified with a Boolean valued topological space (cf. Subsection 5.3.3). We do not claim that this list of sources for many valued topological spaces is complete (cf. Subsection 5.3.1); rather we refer the reader to Part III of this book where we will give various applications of many valued topological techniques.