MCS 332 EXAM 1 INFORMATION

In-class test March 15: closed book and notes

• Meaning and properties of basic set operations: union, intersection, difference, product of two sets
• Arbitrary unions, intersections, and products
• Subset vs. element of
• Be able to answer simple true/false questions on the above.
• Function, injection, surjection, bijection
• (Simple/linear) order relation
• Countable vs. uncountable
• Know how to tell whether some basic sets are countable or not.
• The Axiom of Choice
• The Well-ordering theorem/principle
• Metric, metric space
• Epsilon-ball in a metric space
• Borel-Lebesgue theorem
• Norm on a vector space
• Topology, open sets, topological space
• Neighborhood
• Basis for a topology
• The order topology
• The product topology
• The subspace topology
• Closed sets: definition and characterization
• Limit points
• Closure, closure properties
• Limit of a sequence, convergence
• Hausdorff spaces
• Continuous functions
  • from (a subset of) R to R (the reals)
  • from a metric space X to a metric space Y
  • from a topological space X to a topological space Y
• Homeomorphism
• Product topology
• Box topology

Take-home test due March 18: open book and notes

• metrics
• order topology
• closed sets, closure, and/or Hausdorff spaces
• continuity
• ???