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