Shi studies the topic of fuzzy compactness in L-topological spaces. For complete DeMorgan algebras (L,∨,∧,’) fuzzy compactness is defined as follows. If (X,) is a topological L-space, then G ∈ LX (here LX denotes the set of all L-fuzzy sets on X) is called fuzzy compact. For every family ⊆ of sets from the topology it holds that
where 2() denotes the set of all finite subfamilies of and ≤ is the usual relation on the set {,} where resp. denotes the smallest resp. largest element of LX. In fact, the author mentions that ≤ in the definition above can be replaced by =.
The motivation for studying fuzzy-compact sets comes from the fact that compactness in general topological spaces is a property that is in some sense desirable. Compactness is useful since it is similar to finiteness; that is, some properties that can be proved for finite sets can be proved for compact sets using only slight modifications in the proofs.
Having defined his concept of fuzzy compactness, Shi studies the properties of such fuzzy compact sets. The main contribution of the paper is found in Section 4. Here, some of the properties of fuzzy compact sets are presented. In particular, Theorems 4.1, 4.6, and 4.9 list several equivalent characterizations of fuzzy compact sets.
The paper is written in a strictly formal way, which requires a mathematical background in topology to be fully understood. More examples might have made the proofs and definitions a bit clearer. However, Shi proves that the L-closed interval [a,b](L) is fuzzy compact where the proof of this theorem uses some of the properties derived at earlier stages of the paper. This proof helps readers better understand these properties.
