Alyaa Yousef Khudayer Mutawek

The concepts of paracompact and ideal are two important mathematical notions in topology. They have a significant role in applied science. In light of these two concepts, the idea of merging them into one concept, which we called ID-paracompact space has emerged. Its most important algebraic and topological properties and its relationships to I-paracompact space and paracompact space were studied. In addition, we discussed the effect of some different functions on preserving the ID-paracompact space property.

The idea of the present study is based on the fundamental idea of defining a new set, which we called interior isolated set and the old topological concept of isolated point. Therefore, the topological properties of this concept and its most important relationships were examined. Moreover, we built new types of sets and spaces as follows: IIS*-set, IISopen and TIIS-space. The most important algebraic properties of these sets were also studied. In addition, several new definitions were set, which was called IIS-compact spaces IIS-O-compact. The researcher also discussed their most important properties and relationships with each other.presented for compact based on interior isolated

Given the importance of proximity spaces in scientific and engineering applications, this study attempts to expand the concept of compact spaces in proximity spaces while preserving the general framework of the concept of a compact space. Three types of compact spaces are presented, referred to respectively, δ-compact, σ-compact, B -compact, along with a study of the main properties of each type.

this paper , we introduce the concept of compactly f-closed set. In this work , we have also shown that where X is a T 2-spase,then a subset A of X is compactly f-closed if and only if A is an f-closed set. Moreover , we have studied the concept of compactly f-k-closed set and shown that if the space is T2, then compactly if-closed is equivalent to compactly f-k-closed.

In this paper ,we introduce the definition of semi-coercive function and introduce severalproperties of semi-coercive function