An important class of spaces was introduced by I.A. Bakhtin (under the name
?metric-type?) and independently rediscovered by S. Czerwik (under the name
?b-metric?). Metric-type spaces generalize ?classic? metric spaces by
replacing the triangularity axiom with a more general axiom d(x,z)? k?
(d(x,y)+ d(y,z)) for all x,y,z ? X where k ? 1 is a fixed constant.
Recently R. Saadadi has introduced the fuzzy version of ?metric-type?
spaces. In this paper we consider topological and sequential properties of
such spaces, illustrate them by several examples and prove a certain version
of the Baire Category Theorem.