Abstract We introduce the notions of N-local compactness and N-openess in fuzzy neighbourhood spaces and in fuzzy neighbourhood groups, together with some characterizations for them. In particular, we prove that a fuzzy neighbourhood space is N-locally compact if and only if its α-level spaces, for all 0< α<1, are locally compact. A similar criterion is established for N-open functions. We also introduce the notion of fuzzy absolute value functions on groups. We show that there exists a 1-1 correspondence between invariant fuzzy (probabilistic) pseudo-metrics on groups and fuzzy absolute value functions. We establish theorems on fuzzy (probabilistic) pseudo-metrizability of fuzzy neighbourhood groups. Finally, fuzzy (probabilistic) metrizability of quotient fuzzy neighbourhood groups is also taken into account.