<i>G</i> _𝔫 -methods and <i>G</i> _𝔫 -compactness via nets

Abstract Sequences, or more generally, nets, are important in numerous branches of mathematics, such as analysis, number theory and topology. For the reasons explained in introduction, in some works we move from sequences to nets. Considering limits of sequences in a Hausdorff space X assigns a function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>lim</m:mo> </m:math> {\lim} from the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>c</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {c(X)} of convergent sequences to X . Inspired by this, a G - method , as a generalization of the function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>lim</m:mo> </m:math> {\lim} , is defined by replacing the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>c</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {c(X)} with a certain class of sequences. Originated by such G -methods, the sequential versions of some fundamental properties in topology, such as continuity, compactness, and connectedness, have been recently generalized by G -methods defined for sequences in sets. This paper presents a study of convergent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mi>𝔫</m:mi> </m:msub> </m:math> {G_{\mathfrak{n}}} -methods involving nets rather than sequences as generalizations of G -methods; and then gives some properties and characterizations of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mi>𝔫</m:mi> </m:msub> </m:math> {G_{\mathfrak{n}}} -compactness and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mi>𝔫</m:mi> </m:msub> </m:math> {G_{\mathfrak{n}}} -countably compactness.