On completeness and compactness in fuzzy metric spaces

In this paper we introduce the concepts of weak-completeness and sequentially p-compactness in fuzzy metric spaces in relation to the works given by Miheţ (Fuzzy Sets Syst 158:915–921, 2007) and Gregori et al. (Topol Appl 156:3002–3006, 2009), which are weaker than earlier concepts. We also study the relationship among the concepts and we support our ideas with some illustrative examples. Finally, we consider some fixed point theorems for weak-complete and sequentially p-compact fuzzy metric spaces, respectively. In particular, we give a partial answer to the question posed by Gregori and Sapena (Fuzzy Sets Syst 125:245–252, 2002) in relation to the fuzzy contractive sequences.