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Published online by Cambridge University Press: 05 May 2013
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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