Vol. 11, Issue 9, Part D (2025)

Abstract
In this study, we thoroughly investigated circle maps and lift maps and defined continuous functions for circle maps and their lifts. Some findings regarding the function and its continuity have been demonstrated. Additionally, some findings based on the continuity of the quotient functions are expanded. Additional findings on the composition of circle maps with lift maps and the proof of the continuity of the connected spaces were also made. Set theory-based fundamental conclusions have been included, which could help in understanding common ideas. Detailed research has also been conducted using lift maps and circle maps that combine lift maps.