An investigation on Pythagorean fuzzy $$\mathfrak {F_{p}^*}$$ fraction dense space using Pythagorean fuzzy frames

The concept of frame is a generalisation of the concept of category of topological space open subsets. As a result, each frame acts as an open set in this context and the Pythagorean fuzzy sets is defined as a frame. The primary goal of this research unit is to investigate the behaviour of Pythagorean fuzzy frames. Pythagorean fuzzy [Formula: see text] structure space is defined using Pythagorean fuzzy frames. Pythagorean fuzzy [Formula: see text] closed sets, Pythagorean fuzzy dense set, Pythagorean fuzzy nowhere dense set, Pythagorean fuzzy somewhere dense set is established in order to investigate the Pythagorean fuzzy frames defined in Pythagorean fuzzy [Formula: see text] structure space. Further, Pythagorean fuzzy [Formula: see text] continuous function is explored in this manuscript. Separation axioms of the Pythagorean fuzzy [Formula: see text] structure space is established in order to comprehend the Pythagorean fuzzy frame. Additionally Pythagorean fuzzy [Formula: see text] fraction dense space and Pythagorean fuzzy [Formula: see text] space is defined and explored to examine the behaviour of defined Pythagorean fuzzy frames.