On I-convergence of nets of functions in fuzzy metric spaces

Abstract In this paper, we introduce ideal versions of semi- α convergence, semi-exhaustiveness and semi uniform convergence of nets of functions between two fuzzy metric spaces, and obtain some properties of them. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">I</m:mi> </m:math> $\mathcal{I}$ be a D -admissible ideal on the directed set ( D , ≥), X and Y be two fuzzy metric spaces, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>d</m:mi> <m:mo>∈</m:mo> <m:mi>D</m:mi> </m:mrow> </m:msub> </m:math> ${\left({f}_{d}\right)}_{d\in D}$ be a net of functions in Y X and f ∈ Y X . We mainly show that: (1) If <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>d</m:mi> <m:mo>∈</m:mo> <m:mi>D</m:mi> </m:mrow> </m:msub> </m:math> ${\left({f}_{d}\right)}_{d\in D}$ is a net of continuous functions and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">I</m:mi> </m:math> $\mathcal{I}$ -semi uniformly convergent to f , then f is continuous; (2) The following are equivalent: (a) the net <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>d</m:mi> <m:mo>∈</m:mo> <m:mi>D</m:mi> </m:mrow> </m:msub> </m:math> ${\left({f}_{d}\right)}_{d\in D}$ is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">I</m:mi> </m:math> $\mathcal{I}$ -semi-exhaustive and pointwise <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">I</m:mi> </m:math> $\mathcal{I}$ -convergent to f ; (b) the net <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>d</m:mi>