###### Learning Outcomes:

On successful completion of this module, the student is expected to be able to understand the definitions, theorems and examples covered in all of the topics listed above. These include, but are not limited to, being able to define a topological space and reproduce examples like the discrete, indiscrete and cofinite topologies; give examples of topological spaces that do or do not satisfy the Hausdorff separation axiom; determine the set of accumulation points and the closure of a given subset; understand continuity and its many equivalent definitions; provide examples of continuous functions and homeomorphisms; construct products of finitely many topological spaces and understand continuity in this setting; construct common examples of quotient spaces; define compactness using open covers; show that closed bounded intervals of the real line are compact; show that products of finitely many compact spaces are compact; apply the Heine-Borel Theorem; define and compare connectedness and path connectedness using examples such as the topologist's sine curve; show that intervals of the real line are connected; show that products of finitely many connected spaces are connected; determine the connected components of a topological space; apply Brouwer's Fixed Point Theorem.