The purpose of the present work is to introduce and investigate some of the new topological concepts by using the class of Sp-open sets such as Sp-closed spaces, Sp-separation axioms, Sp-connected and Sp-paracompact spaces.
We introduce and investigate the concept of Sp-convergence and Sp-accumulation
points. It is shown that every filterbase which is Sp-convergent (resp. has an Sp-accumulation point) in the topological space is Sp- -convergent (resp. has the same Sp- -accumulation point) and there is no relation between Sp- -convergence (Sp- -accumulation) for a filter base with s-convergence and -convergence (resp. s-accumulation and -accumulation) points.
Also we introduce the class of Sp-closed space by using Sp-open sets which is a generalization of s-closed spaces. This class is contained in the class of Sc-closed spaces and it contains the class of the s-closed spaces and in an extremally disconnected space the concept of p-closed spaces and Sp-closed spaces are identical, and if the space is locally indiscrete, then both of the Sp-closed spaces and s-closed spaces are equivalent. It is proved that the image of an Sp-closed space under Sp-continuous function is quasi-H-closed.
The concepts of the Sp-Ti spaces are introduced as a generalization of the concept of
strongly semi-Ti spaces for i = 0, 1, 2. Also Sp-regular and Sp-normal space are studied and the relations among them are obtained.
The concept of Sp-connectedness is defined which is a strong form of semi- connectedness and weaker than connectedness. It is showed that the image of Sp-connected space is Sp-connected under a surjective open continuous function.
The concept of Sp-paracompactness is defined and some results on it are obtained and it has been shown that it is stronger than S-paracompactness. It is shown that if the space is locally indiscrete Sp-paracompact space, then it is P3-paracompact and if A is a g-closed subset of an Sp-paracompact space , then A is Sp-paracompact relative to X . Also for the subset A of a Hausdorff Sp-paracompact space, the following are equivalent:
- A is Sp-paracompact relative to X.
- For each x in the complement of A there exists an Sp-open set U such that xÎUÍ Spcl( U ) Í X\A .
- A is θs -closed set.
- A is closed set.