Subhankar Jana - Department of Mathematics, National Institute of Technology Silchar, Silchar, Assam, ۷۸۸۰۱۰, India.
Anjali Patel - Department of Mathematics, National Institute of Technology Silchar, Silchar, Assam, ۷۸۸۰۱۰, India.
Juthika Mahanta - Department of Mathematics, National Institute of Technology Silchar, Silchar, Assam, ۷۸۸۰۱۰, India.

The fuzzy set generally identifies the topological relationship between two vague spatial objects. Indeterminacy can arise at any point in the modelling process, and the fuzzy model is unable to deal with this. Given that the Pythagorean fuzzy is better equipped to deal with indeterminacies than the fuzzy set, we advocated using Pythagorean fuzzy modelling to determine the topological relation between two spatial objects. This paper contributes to the expanding study of Pythagorean fuzzy topological spaces by introducing core, fringe, outer, regular open set, regular closed set, double connectedness and homeomorphism in Pythagorean fuzzy environment. Using these definitions, the paper proposes an algebraic model, namely the pythagorean fuzzy ۹-intersection matrix, to find topological relations between any two Pythagorean fuzzy sets in a Pythagorean topological space. The inbuilt capability of the PFS to handle indeterminacy establishes the proposed model as the potential tool to find topological relations between two indeterminate spatial objects. Ample instances are discussed to nourish the existence of indeterminate spatial objects. Finally, a simple Pythagorean fuzzy region is defined and all possible relations between such two Pythagorean fuzzy regions are examined.

Pythagorean fuzzy set, Pythagorean fuzzy topology, ۹-intersection matrix, GIS, Vague object modelling