Topology, a foundational pillar in modern mathematics, plays a crucial role in understanding the properties of spaces preserved under continuous transformations. This paper delves into the intricate realm of homotopy equivalences within infinite-dimensional topological spaces. The primary objective of this research is to establish new conditions under which two infinite-dimensional spaces can be considered homotopically equivalent. We employ advanced techniques involving simplicial complexes and spectral sequences to dissect the structural nuances of these spaces. Our findings reveal a series of previously uncharted equivalences that expand the current understanding of infinite-dimensional topology. Furthermore, we introduce a novel framework for classifying these spaces based on their homotopic properties, offering a comprehensive perspective that aligns with both theoretical and practical applications. The implications of this research extend to various domains, including algebraic topology and geometric group theory, providing new avenues for exploration and development. We conclude that these insights not only advance the theoretical foundation of topology but also offer practical methodologies applicable in complex systems analysis.