Ideal Convergence In Quasi-metric Spaces And Topological Groups

Ideal convergence is a generalization of statistical convergence.This thesis mainly studies the properties of ideal convergence in quasi-metric spaces and topological groups.In Chapter 2,we first define some ideal versions of Cauchy sequences in quasi-metric spaces,and construct examples to illustrate the implication relationship between these sequences.Then,we study I-sequentially completeness and I-sequentially compactness in quasi-metric spaces,and give a property of I-sequentially complete quasi-metric space with decreasing I-closed sets.Moreover,it is proved that every precompact left Isequentially complete quasi-metric space is compact when I is an P-ideal in N.In Chapter 3,we mainly discuss the ideal convergence of nets in topological groups.Firstly,several operational properties of the ideal convergence of nets in topological groups are explored.It is proved that the ideal convergence of nets satisfies multiplication and reversible operations in topological groups.Moreover,when G is a topological T2 group satisfying the first countable axiom and I is a P-ideal on directional set D,if the net（Xα）α∈D is I-convergent to x in G,then there are nets（yα）α∈D and（zα）α∈D satisfying lim yα=x,xα=yαzα,{α∈D:xα≠yα}∈I and {α∈D:zα≠e}∈I in G.Finally,it is proved in Banach space that if I is a D-admissible ideal,then（1）C（I）is a closed linear subspace of l∞D;（2）（?）=C（I）,where l∞D is the set of all bounded nets in Banach space and endowed the supremum norm,C（I）and C（I*）are the sets of all bounded I-convergent and I*-convergent nets in Banach space,respectively.