A Study on Homotopy Invariance of Circle and Stereographic Projection

This paper explores fundamental applications of topological spaces and stereographic projection derived from the properties of the circle, employing key concepts such as continuous functions and homotopy theory. By examining the behavior of mappings and deformations within topological spaces, the study demonstrates how the circle serves as a foundational structure for understanding more complex topological constructs. Special attention is given to the use of stereographic projection in visualizing the relationship between the circle and the unit sphere, illustrating how these mathematical tools contribute to a deeper understanding of continuity and homotopy in topological analysis. The discussion offers a concise yet insightful introduction to the interplay between geometric intuition and topological formalism.