Second Order Ideal-Ward Continuity

The main aim of the paper is to introduce a concept of second order ideal-ward continuity in the sense that a function is second order ideal-ward continuous if whenever and a concept of second order ideal-ward compactness in the sense that a subset of is second order ideal-ward compact if any sequence of points in has a subsequence of the sequence x such that where . We investigate the impact of changing the definition of convergence of sequences on the structure of ideal-ward continuity in the sense of second order ideal-ward continuity and compactness of sets in the sense of second order ideal-ward compactness and prove related theorems. 1. Introduction Let us start with basic definitions from the literature. Let , the set of all natural numbers, and . Then the natural density of is defined by if the limit exists, where the vertical bars indicate the number of elements in the enclosed set. Fast [1] presented the following definition of statistical convergence for sequences of real numbers. The sequence is said to be statistically convergent to if for every , the set has natural density zero; that is, for each , In this case, we write or and denotes the set of all statistically convergent sequences. Note that every convergent sequence is statistically convergent but not conversely. Some basic properties related to the concept of statistical convergence were studied in [2, 3]. In 1985, Fridy [4] presented the notion of statistically Cauchy sequence and determined that it is equivalent to statistical convergence. Caserta et al. [5] studied statistical convergence in function spaces, while Caserta and Ko inac [6] investigated statistical exhaustiveness. Kostyrko et al. [7] introduced the notion of ideal convergence. It is a generalization of statistical convergence. For details on ideal convergence we refer to [8–13]. Let be a nonempty set; then a family of sets (power sets of ) is called an ideal on if and only if(a) , (b)for each , we have ,(c)for each and each , we have . A nonempty family of sets is a filter on if and only if(a) , (b)for each , we have ,(c)each and each , we have . An ideal is called nontrivial ideal if and . Clearly is a nontrivial ideal if and only if is a filter on . A nontrivial ideal is called admissible if and only if . A nontrivial ideal is maximal if there cannot exist any nontrivial ideal containing as a subset. Definition 1 (see [7]). A sequence of points in is said to be -convergent to the number if, for every , the set . One writes . One sees that a sequence being -convergent implies that . Burton and Coleman [14]