Abstract:

 

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:

  1. A is Sp-paracompact relative to X.
  2. For each x in the complement of A there exists an Sp-open set U such that xÎUÍ Spcl( U ) Í X\A              .
  3. A is  θs -closed set.
  4. A is closed set.

 

 

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