Math 373

          Topological spaces: Definition and examples of topological spaces, Comparison between topological spaces, Open and closed sets, Closure of a set, Interior, boundary , exterior  and derived sets,  Subspaces. Basis: Definition and examples, The generated topology on a set, Comparison between topologies using basis, Equivalence of basis. Product topology: Finite product topology, Interior, Boundary and Closure of the product of two sets, Examples. Subbasis. Metric spaces: Definition and examples of the Metric, Discrete metric, Usual and square metric on Rn, Metric topology , Metrizabilty, Hausdorff space, Sequences in topological spaces, Uniqueness of a sequence limit, Sequence lemma, The usual space Rn is metrizable, Continuity: Definition of continuous functions and examples, Characterization of continuous functions in topological spaces and metric spaces, Homeomorphisms, open and closed functions, Examples, Topological property, Examples of homeomorphisms. Compactness: Definitions and example of a cover , Open cover ,  Subcover and Finite subcover, Definition and examples of compact spaces, Some properties of compactness, Compactness in Rn, Limit point compactness, Sequentially compact spaces, Compactness in metric spaces, Finite intersection property, Compactness and finite intersection property.