Abstract. We introduce an approach to the concept of bornology in the framework of many-valued mathematical structures and develop the basics of the theory of many-valued bornological spaces and initiate the study of the category of many-valued bornological spaces and appropriately defined bounded "mappings" of such spaces. A scheme for constructing many-valued bornologies with prescribed properties is worked out. In particular, this scheme allows to extend an ordinary bornology of a metric space to a many-valued bornology on it.In the academic year 1978/79 I was lucky to get a post-doc research position at the University of Zagreb and to spend this time working in collaboration with professor Sibe Mardešić in shape theory. It was a real pleasure to work together and to communicate generally with this outstanding mathematician, talented teacher and a very kind and nice man. Unfortunately, at present I am quite far from shape theory. Therefore the paper I would like to dedicate to Sibe Mardešić, written together with my former PhD student Ingrīda Uljane, is in the field of so called many-valued mathematical structures -the area of my main mathematical interests at present.

No abstract

No abstract

We present a many-level version for the Pawlak -Dubois&Prade theory of rough approximation of fuzzy sets. Basing on the many-level upper and lower fuzzy rough approximation operators, we define the measure of rough approximation that in a certain sense characterizes the quality of the obtained approximation. Further, the fuzzy rough approximation operators give rise to two alternative topological-type structures considered in the paper. Keywords: Many-level fuzzy rough approximation operators, measure of fuzzy rough approximation, LM -fuzzy (di)topologies, Mlevel L-fuzzy (di)-topologies.2 Many-level L-fuzzy relations 2.1 Basic definitions Let L = (L, ≤ L , ∧ L , ∨ L , * ) be an integral commutative complete lattice monoid (in particular, L = [0, 1] and * a lower semi-continuous t-norm), see, e.g. [8], and let M = (M, ≤ M , ∧ M , ∨ M ) be a complete infinitely distributive lattice. Definition 2.1 An M -level L-fuzzy relation on a set 11th

After the inception of the concept of a fuzzy metric by I. Kramosil and J. Michalek, and especially after its revision by A. George and G. Veeramani, the attention of many researches was attracted to the topology induced by a fuzzy metric. In most of the works devoted to this subject the resulting topology is an ordinary, that is a crisp one. Recently some researchers showed interest in the fuzzy-type topologies induced by fuzzy metrics. In particular, in the paper (J. J. Miñana, A. Sostak, Fuzzifying topology induced by a strong fuzzy metric, Fuzzy Sets and Systems 300 (2016), 24-39) a fuzzifying topology T : 2 X → [0, 1] induced by a fuzzy metric m : X × X × [0, ∞) was constructed. In this paper we extend this construction to get the fuzzy topology T : [0,1] X → [0, 1] and study some properties of this fuzzy topology.2010 MSC: 54A40.