The 680 Series |

**The basics of topology at Ohio
University are covered in the 680 series. These courses form one of the core
areas in which students are allowed to take comprehensive exams. For more
information on the courses, feel free to contact a member of our
topology group.**

**680A is essentially an
introductory course on set theory, generally covering the following topics**

**relations on sets****equivalence relations****partially and linearly ordered sets****functions/mappings****injections, surjections, bijections****chains of sets****Zorn's Lemma and its equivalents (Axiom of Choice, Zermelo's Theorem)****Filters, filter bases, and ultrafilters****Countable and uncountable sets****cardinal numbers and basic cardinal arithmetic****well-ordered sets, ordinals, and transfinite induction****the idea of axiomatic set theory****the idea of undecidable statements**

**680B covers the basic
topological notions and theorems, including the following topics**

**metric spaces and topological spaces****open sets, closed sets, closures, and boundaries****bases, dense sets, and networks****complete metric spaces and totally bounded metric spaces****the Baire Category Theorem for complete metric spaces****every contraction of a complete metric space has a fixed point****completion of a metric space****topologies generated by orderings, especially linear orderings****separation axioms****metrizable and non-metrizable spaces****Urysohn metrization theorem****every open covering of a separable metrizable space has a countable subcovering****continuous functions, quotient spaces, and quotient mappings****homeomorphisms****Peano curves****Completely regular spaces and normal spaces****Urysohn's Lemma and the Tietze Extension Theorem****Compactness and local compactness****characterizations of compactness in various classes of spaces****preservation of compactness by continuous maps****Baire Category Theorem for compact Hausdorff spaces****one-point compactifications of locally compact spaces****the Čech-Stone compactification of a completely regular space****Tychonov's Theorem****Connected and locally connected spaces****The notion of a topological manifold**

**680C includes more
specialized topics, often determined by the instructor. Examples of
potential topics are**

**metrization and paracompactness****introduction to dimension theory****Sperner's Lemma and Brouwer's Fixed-point Theorem****homotopy and the classification of self-maps of the circle****topological groups and examples of Lie groups****the fundamental group of a topological space**