On Baire approximations of normal integrands
In this paper we introduce the notation of t-best approximatively compact sets, t-best approximation points, t-proximinal sets, t-boundedly compact sets and t-best proximity pair in fuzzy metric spaces. The results derived in this paper are more general than the corresponding results of metric spaces, fuzzy metric spaces, fuzzy normed spaces and probabilistic metric spaces.
The structure of binary coproducts in the category of frames is analyzed, and the results are then applied widely in the study of compactness, local compactness (continuous frames), separatedness, pushouts and closed frame homomorphisms.
We show that for each natural number n > 1, it is consistent that there is a compact Hausdorff totally disconnected space such that has no uncountable (semi)biorthogonal sequence where ’s are atomic measures with supports consisting of at most 2n-1 points of , but has biorthogonal systems where ’s are atomic measures with supports consisting of 2n points. This complements a result of Todorcevic which implies that it is consistent that such spaces do not exist: he proves that its is...