Randall, Jessica2025-10-022025-10-022024https://hdl.handle.net/10566/21008This thesis addresses a key challenge in topology: determining whether spaces are homeomorphic, which requires establishing a continuous, invertible mapping. Traditional methods based on basic invariants like connectivity and compactness often fall short for complex classifications. To address this, we draw on advanced concepts from algebraic topology, particularly the fundamental and homology groups introduced by Henri Poincar ́e and Enrico Betti, as well as Betti numbers that quantify the dimensions of “ holes ” in spaces. We analyze spatial structures through homology by focusing on missing elements. The p-th homology group Hp enables comparisons between the p-th cycle group Zp and the p-th boundary group Bp, allowing us to isolate sig- nificant topological features.enHomologyTopological DataAstronomical DataSpatial DistributionsHenri Poincar ́eThesis