It is shown that if a subset of a topological space (χ, τ) is δ-semi.closed, then it is semi.closed. By use this fact, we introduce the concept regularity of a topological space (χ, τ) via δ-semi.open sets. Many properties and results were investigated and studied. In addition we study some maps that preserve the δ-semi.regularity of spaces.

Let be a commutative ring with unity and let be a non-zero unitary module. In
this work we present a -small projective module concept as a generalization of small
projective. Also we generalize some properties of small epimorphism to δ-small
epimorphism. We also introduce the notation of δ-small hereditary modules and δ-small
projective covers.

In this paper we study the concepts of δ-small M-projective module and δ-small M-pseudo projective Modules as a generalization of M-projective module and M-Pseudo Projective respectively and give some results.

In this paper we study the concepts of δ-small M-projective module and δ-small M-pseudo projective Modules as a generalization of M-projective module and M-Pseudo Projective respectively and give some results.

In the present paper, new concepts of generalized continuous mappings, namely Е_c and δ-ß_c-continuous mappings have been introduced and studied by using a new generalized of open sets Е_c and δ-ß_c-open sets ,respectively. Several characterizations and fundamental properties of these forms of generalized continuous mappings are obtained. Moreover, the graphs of Е_c-continuous and δ-ß_c-continuous mappings have been investigated. In addition, the relationships among Е_c-continuous and δ-ß_c-continuous mappings and other well-known forms of g