Sets of Statistical Cluster Points and 𝓘-Cluster Points

Statistical convergence in topology

“…Entourages of the diagonal are supposed to be open and symmetric. Several results about statistical convergence on the real line and in metric spaces (see, for example, [4,33,34]) are true also in uniform spaces.…”

“…(1) (2 X , τ Δ + ) satisfies α 2 (s-Σ E , Σ E ); (2) (2 X , τ Δ + ) satisfies α 3 (s-Σ E , Σ E ); (3) (2 X , τ Δ + ) satisfies α 4 (s-Σ E , Σ E ); (4) (2 X , τ Δ + ) satisfies S 1 (s-Σ E , Σ E ). (3) ⇒ (4).…”

Effects on rough I-lacunary statistical convergence to induce the weighted sequence

“…A sequence x = {x n } n∈N in a normed linear space ||.||, is said to be rough I-convergent to x * w.r.t. the roughness of degreer (or shortly:r-I-convergent to x * ) if for every ε > 0, the set On further progress, we combine the approaches of I-lacunary statistical convergence [10], rough Iconvergence [13,26], statistical cluster point [3,27] and weighted lacunary statistical convergence [21] and introduce new and more advance summability methods namely, rough weighted I-lacunary statistical limit set and weighted I-lacunary statistical cluster points set of a sequence in a metric space. Some new examples are constructed to ensure the deviation from basic results such as: Theorem 1.7 [13,26].…”

The set of filter cluster functions

“…Some applications of this convergence were studied on the asymptotic behavior of optimal paths, variational and optimal control problems in discrete time (see [28], [29], [33], [34], [38]). Then Kostyrko et al ([24]) defined the concepts of I-limit point and I-cluster point which are the generalizations of statistical ones (see also [11], [26]).…”