Statistical convergence in topology

Abstract

We introduce and investigate statistical convergence in topological and uniform spaces and show how this convergence can be applied to selection principles theory, function spaces and hyperspaces.

On some consequences of a generalized continuity

“…It can be noted that summability methods like Cesaro summability, statistical (ideal) convergence (see [14], [13]) and more can be defined in topological spaces and so in general normed linear spaces. As most of these summability methods give rise to regular methods considered here, so the above modification of the notion of continuity to generalized continuity seems very natural.…”

A variation on lacunary statistical quasi cauchy sequences

“…This is denoted by st lim k = L (see [21], [22], [26], [17], and [27]). ( k ) is statistically quasi-Cauchy, or st-quasi-Cauchy if ( k ) is a st-null sequence, where k = k+1 k for each integer n in N, the set of positive integers ( [9]).…”

Statistical convergence and measure convergence generated by a single statistical measure

“…In the recent years, various kinds of statistical convergence and their applications were extensively discussed in many pure and applied mathematical fields. As an example among a large amount of literature, we refer the reader to [3,15,16,22,23,25,26,30,36,37,42], and [49]. A sequence (x n ) in a topological space X is said to be statistically convergent to x ∈ X provided for any neighborhood U of x, we have χ A(ε) (j) = 0 ∞ (I) ⊂ ∞ and the quotient space ∞ / ∞ (I) for an ideal I.…”