Abstract
Sequence converges is an important research object in topology and analysis, since it is closely related to continuity, compactness and other related properties. In this article, we use the notion of regularly convergence, which is a generalization of the convergence notion, to define the regularly sequentially closed sets and the operator of regularly sequential closure; then we consider thier charactarizations and prove that the convergence and regularly convergence are coincide in regular spaces. Finally we introduce new axioms by involving δ-open sets and regular open sets with the concept of regularly convergence; namely δ-sequential space and r-sequential space, when we show that there are no general relations between these new spaces and the sequential space, in addition we prove some statements as; r-first countable space is δ-sequential, r-sequential space is stronger than δ-sequential space and the spaces δ-sequential and sequential are coincide both in regular spaces and in compact r-T2 spaces.