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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