### On fuzzy k-pseudometrics and fuzzy k-pseudometric spaces

#### Abstract

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 \lq\lq b-metric").

Metric-type spaces generalize "classic" metric spaces by replacing the triangularity axiom with a more general axiom

$d(x,z) \leq k\cdot(d(x,y)+d(y,z))$ for all $x,y,z \in X$ where $k \geq 1$ is a fixed constant.

In this paper, we introduce a fuzzy version of a metric-type space calling it a fuzzy k-(pseudo)metric space, illustrate it by several examples,

and study topological properties of such spaces.

We consider sequences in fuzzy k-(pseudo)metric spaces, define the property of completeness

for such spaces and

prove a certain version of the Baire Category Theorem.

(under the name \lq\lq b-metric").

Metric-type spaces generalize "classic" metric spaces by replacing the triangularity axiom with a more general axiom

$d(x,z) \leq k\cdot(d(x,y)+d(y,z))$ for all $x,y,z \in X$ where $k \geq 1$ is a fixed constant.

In this paper, we introduce a fuzzy version of a metric-type space calling it a fuzzy k-(pseudo)metric space, illustrate it by several examples,

and study topological properties of such spaces.

We consider sequences in fuzzy k-(pseudo)metric spaces, define the property of completeness

for such spaces and

prove a certain version of the Baire Category Theorem.

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