Test 3 Study Guide

This test will be cummulative, so review the test 1 study guide and test 2 study guide as well as both tests. In addition:

Additional Definitions

  • Cover
  • Compactness
  • The statement of Heine-Borel
  • Be sure to know all the definitions on the study guides from tests 1 and 2 [many of these should hopefully feel very familiar by now]

    Additional Examples

  • Examples of finite and infinite topological spaces that are compact and why.
  • Examples of topologicial spaces that are not compact and why.
  • An example of a bounded set in a metric space that is not compact and why.
  • An example of a closed set in a metric space that is not compact and why.
  • Compactness in the Cantor set
  • Compactness in the discrete topology
  • Compactness in the cofinite topology
  • Compactness of subsets of R - be able to produce covers that have no finite subcovers
  • All the surfaces that locally look like the plane.
  • Be sure to know all the examples on the study guides from tests 1 and 2 [many of these should hopefully feel very familiar by now]

    Additional Proofs

  • Prove that an infinite discrete topology is not compact
  • Prove that a set with the cofinite topology is compact
  • Prove that a continuous surjection preserves compactness
  • Be sure to know all the proofs on the study guides from tests 1 and 2 [there were 3 proofs on each study guide]

    Big Picture

  • Why topology was important historically
  • Why a notion like compactness is important
  • How did we prove that [0,1] is compact [big picture in one or two sentences - not the entire proof]