Many Valued Topological Spaces

Ulrich Höhle
Additional contact information
Ulrich Höhle: Bergische Universität, Fachbereich Mathematik

Chapter Chapter 5 in Many Valued Topology and its Applications, 2001, pp 145-204 from Springer

Abstract: Abstract 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.

Keywords: Topological Space; Boolean Algebra; Complete Lattice; Image Filter; Borel Subset (search for similar items in EconPapers)
Date: 2001
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4615-1617-0_6

Ordering information: This item can be ordered from
http://www.springer.com/9781461516170

DOI: 10.1007/978-1-4615-1617-0_6

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().