Explore the extension of persistent homology to infinite CW complexes in this 45-minute talk from the Applied Algebraic Topology Network. Delve into a theoretical framework that broadens the application of persistent homology beyond finite complexes. Discover how the Bottleneck Stability Theorem and the Algebraic Decomposition Theorem remain valid without finiteness assumptions. Gain insights into new possibilities for applying persistent homology in topological data analysis, opening up avenues for tackling a wider range of problems in the field.