Explore the fourth lecture in a graduate course on O-minimality and Applications, delivered by Jacob Tsimerman from the University of Toronto at the Fields Institute. Delve into advanced mathematical concepts such as infinitesimal thickening, topological spaces, definable spaces, and definable cell decomposition. Examine the intricacies of Abelian shields, push forward factors, and finite covering. Gain a comprehensive understanding of these complex topics through a structured syllabus that progresses from introductory concepts to a detailed proof and summary of cell decomposition theory.