| Summary: | Recently, the notion of a classical <i>k</i>-metric, which make the triangle inequality to a more general axiom: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>)</mo><mo>≤</mo><mi>k</mi><mo>(</mo><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>+</mo><mi>d</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>, has been presented and is applied in many fields. In this paper, the definitions of an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>L</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></semantics></math></inline-formula>-fuzzy <i>k</i>-pseudo metric and an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>L</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></semantics></math></inline-formula>-fuzzy <i>k</i>-remote neighborhood ball system are introduced. It is proved that the category of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>L</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></semantics></math></inline-formula>-fuzzy <i>k</i>-pseudo metric spaces is isomorphic to the category of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>L</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></semantics></math></inline-formula>-fuzzy <i>k</i>-remote neighborhood ball spaces. Besides, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>L</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></semantics></math></inline-formula>-fuzzy topological structures induced by an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>L</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></semantics></math></inline-formula>-fuzzy <i>k</i>-pseudo metric are presented and their properties are investigated. Finally, the concept of a nest of pointwise <i>k</i>-pseudo metrics is proposed and it is shown that there is a one-to-one correspondence between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>L</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></semantics></math></inline-formula>-fuzzy <i>k</i>-pseudo metrics and nests of pointwise <i>k</i>-pseudo metrics.
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