1 Open Sets Topological space, open sets 2 Bases of a Topological Space Bases, subbases, and local bases of a topological space 3 Interior, Closure, and Boundary Basic concepts in topology 4 Other Definitions of Topological Spaces Definitions of topology using closed sets, closure, and neighborhood filters 5 Continuous Functions Properties of continuous functions 6 Initial and Final Topology Initial/final topology and their examples 7 Subspaces Properties of subspaces 8 Presheaves The gluing lemma and the definition of a presheaf 9 Sheaves Sheaves defined on a topological space 10 Quotient Spaces Properties of Subspaces 11 Product Spaces Properties of product spaces 12 Open Mappings and Closed Mappings Definitions of open maps and closed maps, and their relationship to quotient maps 13 Hausdorff Spaces Convergence of sequences and the Hausdorff axiom 14 Compact Spaces Compact spaces, defined by the existence of a finite subcover for every open cover 15 Compactness and Convergence of Filters Characterizing compactness via filter convergence 16 Compactness and Paracompactness Tychonoff theorem, paracompactness, and partitions of unity 17 Proper Maps The relationship between proper maps as universally closed maps and compactness 18 Dimension Definitions of covering dimension and Krull dimension for algebraic geometry 19 Connected Spaces Connected spaces, path-connectedness, and connected components