On the set function

Key Points

• The aim is to introduce a new set function, $ ext{wp}$, and explore its properties related to homogeneous and Hausdorff continua.
• Developed a new set function inspired by earlier works on homogeneous metric continua.
• Examined the implications of this function within the framework of Hausdorff continua with Effros property.
• Analyzed properties and behaviors of the set function $ ext{wp}$.
• Established foundational properties of the set function $ ext{wp}$.
• Demonstrated connections between $ ext{wp}$ and existing concepts in metric continua.
• Identified potential applications of $ ext{wp}$ in further mathematical research.