On nearly compact topological and fuzzy topological spaces

In this paper we introduce the notions of nearly Ω-compact, nearly (α,β)-compact topological and fuzzy topological spaces, nearly bounded, nearly Ω-bounded, nearly (α,β)-bounded sets and fuzzy sets. By examining these aforementioned notions in the present paper the following applications take place: (α) the characterization of nearly compact topological and fuzzy topological spaces through the contribution of weakly θ-upper limit and fuzzy weakly θ-upper limit (see D.N. Georgiou and B.K. Papadopoulos [Fuzzy Sets Syst. 116 (2000) 385–399; Sci. Math. Jpn. 53 (2001) 233–246]) and (β) the characterization of the nearly bounded sets and fuzzy sets through the assistance of weakly θ-upper limit and fuzzy weakly θ-upper limit. These results generalize basic results of S.D. Iliadis and B.K. Papadopoulos [Publ. Math. Debrecen 47 (1–2) (1995) 139–150] and D.N. Georgiou and B.K. Papadopoulos [Fuzzy Sets and Systems 101 (3) (1999) 495–504; J. Fuzzy Math. 6 (4) (1998) 941–955] for nearly compact topological and fuzzy topological spaces.