# An investigation on Pythagorean fuzzy [image] fraction dense space using Pythagorean fuzzy frames

## TL;DR

This paper explores Pythagorean fuzzy frames and their properties in a specialized mathematical space.

## Contribution

The paper introduces and investigates new concepts like Pythagorean fuzzy fraction dense space and continuous functions in this context.

## Key findings

- Pythagorean fuzzy frames are defined and analyzed in a structure space.
- New types of Pythagorean fuzzy sets and separation axioms are established.
- Fraction dense and P* spaces are explored to understand frame behavior.

## Abstract

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 \documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {F_{p}^*}$$\end{document} structure space is defined using Pythagorean fuzzy frames. Pythagorean fuzzy \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G^*}$$\end{document} 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 \documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {F_{p}^*}$$\end{document} structure space. Further, Pythagorean fuzzy \documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {F_{p}^*}$$\end{document} continuous function is explored in this manuscript. Separation axioms of the Pythagorean fuzzy \documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {F_{p}^*}$$\end{document} structure space is established in order to comprehend the Pythagorean fuzzy frame. Additionally Pythagorean fuzzy \documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {F_{p}^*}$$\end{document} fraction dense space and Pythagorean fuzzy \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P^*}$$\end{document} space is defined and explored to examine the behaviour of defined Pythagorean fuzzy frames.