Article ID: MTJPAM-D-20-00046

Title: Characterizations of Generalized Topologically Open Sets in Relator Spaces


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00046; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 39-94

Document Type: Research Paper

Author(s): Themistocles M. Rassias a , Muwafaq M. Salih b , Árpád Száz c

aDepartment of Mathematics, National Technical University of Athens, Athens, Greece

bDepartment of Mathematics, University of Debrecen, Debrecen, Hungary

cDepartment of Mathematics, University of Debrecen, Debrecen, Hungary

Received: 7 December 2020, Accepted: 2 January 2021, Published: 25 April 2021.

Corresponding Author: Árpád Száz (Email address: szaz@science.unideb.hu)

Full Text: PDF

Abstract

A family  \mathcal{R}  of binary relations on a set  X  will be called a relator on  X, and the ordered pair  X{\hskip 0.2 mm}({\hskip 0.2 mm}{\mathcal{R}}{\hskip 0.2 mm})=({\hskip 0.2 mm}X{\hskip 0.2 mm}, \,{\mathcal{R}}{\hskip 0.2 mm})  will be called a relator space.
Each generalized topology  \mathcal{T}  on  X  can be easily derived from the family  \mathcal{R}_{\mathcal{T}}  of all Pervin’s preorder relations  R_{\,V}={\hskip 0.2 mm}V^{{\hskip 0.2 mm}2}{\hskip 0.2 mm}\cup \,V^{{\hskip 0.2 mm}c}\!\times X with  V\in {\mathcal{T}}{\hskip 0.2 mm}.
For a subset  A  of the relator space  X{\hskip 0.2 mm}({\mathcal{R}}),  we may briefly define

\textstyle A^{-}=\text{cl}_{\mathcal{R}}(A)=\bigcap \ \bigl\{\,R^{-1}[A]: \ \ R\in \mathcal{R}\bigr\},
A^{\circ}=\text{int}_{{\mathcal{R}}}(A)=\text{cl}_{{\mathcal{R}}}(A^{c})^{c}  and  A^{\dagger}=\text{res}_{{\mathcal{R}}}(A)=\text{cl}_{{\mathcal{R}}}(A)\setminus A.
Now, we may also naturally define   {\mathcal{T}}_{{\mathcal{R}}}=\{\,A\subseteq X: \ \ A\subseteq A^{{\hskip 0.2 mm}\circ}{\hskip 0.2 mm}\},
\textstyle {\mathcal{D}}_{{\mathcal{R}}}=\bigl\{\,A\subseteq X: \,\ \ A^{{\hskip 0.2 mm}-}={\hskip 0.2 mm}X\,\bigr\} \ \quad\textrm{and}\quad \ {\mathcal{N}}_{{\mathcal{R}}}=\bigl\{{\hskip 0.2 mm}A\subseteq X: \,\ \ A^{{\hskip 0.2 mm}-{\hskip 0.2 mm}\circ}={\hskip 0.2 mm}\emptyset\,\bigl\}\,.
Moreover, following some basic definitions in topological spaces, a subset  A  of the relator space  X{\hskip 0.2 mm}({\hskip 0.2 mm}{\mathcal{R}}{\hskip 0.2 mm})  may, for instance, be naturally called topologically
(1)  regular open  if  A=A^{{\hskip 0.2 mm}-{\hskip 0.2 mm}\circ};
(2)  preopen  if  A\subseteq A^{-\circ};       (3)  semi-open  if  A\subseteq A^{\circ-};
(4)  \alpha–open  if  A\subseteq A^{{\hskip 0.2 mm}\circ{\hskip 0.2 mm}-\circ};       (5)  \beta–open  if  A\subseteq A^{{\hskip 0.2 mm}-\,\circ\,-}{\hskip 0.2 mm};
(6)  quasi-open if there exists  V \in \mathcal{T}_{\mathcal{R}}  such that  V \subseteq A\subseteq V^{{\hskip 0.2 mm}-};
(7)  pseudo-open if there exists  V\in \mathcal{T}_{\mathcal{R}}  such that  A\subseteq V\subseteq A^{{\hskip 0.2 mm}-}.
And, the family of all such subsets A of X(\mathcal{R}) may, for instance, be naturally denoted by {\hskip 0.2mm}{\mathcal{T}}_{{\mathcal{R}}}^{\kappa} with {\hskip 0.2 mm}\kappa=r,   p,  s,  \alpha,  \beta,  q  and   ps,  respectively.
Here, we shall mainly be interested in the relationships and characterizations of the families \mathcal{T}_{\mathcal{R}}^{\kappa}. For instance, we shall prove the following assertions:
(1)  If  {\mathcal{R}}{\hskip 0.2 mm}  is topological, then  {\hskip 0.2 mm}{\mathcal{T}}_{{\mathcal{R}}}^{s}={\hskip 0.2 mm}{\mathcal{T}}_{{\mathcal{R}}}^{{\hskip 0.2 mm}q}. Moreover,  A \in \mathcal{T}_{\mathcal{R}}^{s}  if and only if  {\hskip 0.2 mm}A=V\cup B{\hskip 0.2 mm}  for some  V \in \mathcal{T}_{\mathcal{R}}  and  B\subseteq V^{{\hskip 0.2 mm}\dagger}.
(2)  If  \mathcal{R}  is topological, then  {\mathcal{T}}_{{\mathcal{R}}}^{p}={\mathcal{T}}_{{\mathcal{R}}}^{ps}. Moreover, if in addition  \mathcal{R}  is topologically filtered, then  A\in {\mathcal{T}}_{{\mathcal{R}}}^{{\hskip 0.2 mm}p} if and only if  A=V \cap B   for some  V \in \mathcal{T}_{\mathcal{R}}   and  B\in \mathcal{D}_{\mathcal{R}};
(3)  If  \mathcal{R}  is topological, then  {\mathcal{T}}_{{\mathcal{R}}}^{{\hskip 0.2 mm}\alpha}={\hskip 0.2 mm}{\mathcal{T}}_{{\mathcal{R}}}^{{\hskip 0.2 mm}s}\cap{\hskip 0.2 mm}{\mathcal{T}}_{{\mathcal{R}}}^{{\hskip 0.2 mm}p}. Moreover, if in addition  \mathcal{R}  is topologically filtered, then  A\in {\mathcal{T}}_{{\mathcal{R}}}^{{\hskip 0.2 mm}\alpha}   if and only if  A=V \setminus B  for some  V \in \mathcal{T}_{\mathcal{R}}  and  B \in \mathcal{N}_{\mathcal{R}}.
Set-theoretic properties and some proximal and paratopological counterparts of the families  {\hskip 0.2 mm}{\mathcal{T}}_{{\mathcal{R}}}^{\kappa}{\hskip 0.2 mm}  will be investigated in some subsequent papers.

Keywords: Generalized uniformities, Interiors and closures, Generalized open sets, Characterization theorems

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