Book contents
- Frontmatter
- Contents
- CONTRIBUTORS
- NOTES
- Obituary: Clifford Hugh Dowker
- Knot tabulations and related topics
- How general is a generalized space?
- A survey of metrization theory
- Some thoughts on lattice valued functions and relations
- General topology over a base
- K-Dowker spaces
- Graduation and dimension in locales
- A geometrical approach to degree theory and the Leray-Schauder index
- On dimension theory
- An equivariant theory of retracts
- P-embedding, LCn spaces and the homotopy extension property
- Special group automorphisms and special self-homotopy equivalences
- Rational homotopy and torus actions
- Remarks on stars and independent sets
- Compact and compact Hausdorff
- T1 - and T2 axioms for frames
Some thoughts on lattice valued functions and relations
Published online by Cambridge University Press: 05 May 2013
- Frontmatter
- Contents
- CONTRIBUTORS
- NOTES
- Obituary: Clifford Hugh Dowker
- Knot tabulations and related topics
- How general is a generalized space?
- A survey of metrization theory
- Some thoughts on lattice valued functions and relations
- General topology over a base
- K-Dowker spaces
- Graduation and dimension in locales
- A geometrical approach to degree theory and the Leray-Schauder index
- On dimension theory
- An equivariant theory of retracts
- P-embedding, LCn spaces and the homotopy extension property
- Special group automorphisms and special self-homotopy equivalences
- Rational homotopy and torus actions
- Remarks on stars and independent sets
- Compact and compact Hausdorff
- T1 - and T2 axioms for frames
Summary
INTRODUCTION
A lattice valued function is a function f: X → L from a set X to a lattice L. Both X and L may possess further structure. In fact, every real-valued function is lattice-valued by virtue of the usual max, min lattice on the ordered set of reals. It could not be our intention to discuss such a general situation. We concentrate, rather, on some areas where the actual lattice structure of L plays a major part in a topological theory. Continuous real-valued functions from a topological space are thus excluded per se, but feature within the context of fuzzy topological spaces. Some of the formal transition from ordinary to fuzzy spaces is largely mechanical. More interesting are the difficulties encountered, and on some of these we shall concentrate.
The term ‘fuzzy’ has been used by Poston (22) and Dodson (6) to describe a set with a reflexive, symmetric relation, elsewhere (30) called a tolerance space. We shall avoid confusion by adopting the latter term. Extending the tolerance relation to a fuzzy, (or L-fuzzy, Goguen (8)) relation, we review the topological analogues which can be introduced, with particular reference to homogeneity.
The main discussion, then, is on two topics, namely fuzzy topological spaces and sets with fuzzy relations. We conclude with a few general remarks on lattice-valued functions, topology and homology.
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- Aspects of TopologyIn Memory of Hugh Dowker 1912–1982, pp. 127 - 140Publisher: Cambridge University PressPrint publication year: 1985