mati Abel - Institute of Pure Mathematics, University of Tartu, J.Liivi street ۲, EE-۵۰۴۰۹ Tartu, Estonia
aleksandrs ˇSostaks - Department of Mathematics, University of Latvia, Zellu street ۸, LV-۱۰۰۲, Riga, Latvia and Institute of Mathematics and CS, University of Latvia, Raina bulv. ۲۹, LV-۱۵۸۶, Riga, Latvia

The concept of an L-bornology is introduced and the theory of L-bornological spacesis being developed. In particular the lattice of all L-bornologies on a given set is studied and basic properties ofthe category of L-bornological spaces and bounded mappings are investigated.

Bornology, L-set, L-bornology, Fuzzy set, fuzzy topology