In this work, the notion is defined by using    and some properties of this set are studied also,  and   Ù€ set are two concepts that are defined by using ; many examples have been cited to indicate that the reverse of the propositions and remarks is not achieved. In addition, new application example of nano was studied.

Soft closure spaces are a new structure that was introduced very recently. These new spaces are based on the notion of soft closure operators. This work aims to provide applications of soft closure operators. We introduce the concept of soft continuous mappings and soft closed (resp. open) mappings, support them with examples, and investigate some of their properties.

In this paper, the terms of Lascoux and boundary maps for the skew-partition (11,7,5) / (1,1,1) are found by using the Jacobi-Trudi matrix of partition. Further, Lascoux resolution is studied by using a mapping Cone without depending on the characteristic-free resolution of the Weyl module for the same skew-partition.

In this paper, the complex of Lascoux in the case of partition (3,3,2) has been studied by using diagrams ,divided power of the place polarization ) (k ij ,Capelli identites and the idea of mapping Cone .

The main aim of this paper is to study the application of Weyl module resolution in the case of two rows, which will be specified in the skew- partition (6, 6)/(1,1) and (6,6)/(1,0), by using the homological Weyl (i.e. the contracting homotopy and place polarization).

     The aim of this work is to survey the two rows resolution of Weyl module and locate the terms and the exactness of the Weyl Resolution in the case of skew-shape (8,6)/(2,1).

The aim of this work is to study the application of Weyl module resolution in the case of two rows, which will be specified in the partition (7, 6) and skew- partition (7,6)/(1,0)  by using the homological Weyl (i.e. the contracting homotopy and place polarization).

The purpose of this paper is to study the application of Weyl module’s resolution in the case of two rows which will be specified in the partitions (7, 7) and (7, 7) / (1, 0), using the homological Weyl (i.e. the contracting homotopy and place polarization).

By use the notions pre-g-closedness and pre-g-openness we have generalized a class of separation axioms in topological spaces. In particular, we presented in this paper new types of regulαrities, which we named ρg­regulαrity and Sρg­regulαrity. Many results and properties of both types have been investigated and have illustrated by examples.