Test 1 on material including the exercise sets up through those on topologies. Review definitions, examples, reasons why statements are true or false and proofs and "how to's".

In order to help you further learn the material, which we will continue using, this test will be closed to notes.

Definitions

• Set containment
• Function
• Inverse image of a function
• Intersection
• Union
• Complement
• Continuity in ℜ
• Metric space
• Continuity in a metric space
• Metric ball
• Open set in a metric space
• Topology
• Open in a topology
• Comparing topologies

Examples

• Discrete topology and the metric it arises from
• Standard topology on ℜ and a metric it arises from
• Square metric
• Taxicab metric
• Standard topology on ℜ2 and two different metrics it arises from
• Sierpinski space
• Cofinite topology
• Lower limit topology
• Open sets in the standard toplogy on ℜ (and sets that are not open) and an explanation of why
• Open sets in the cofinite topology on ℜ (and sets that are not open) and an explanation of why
• Open sets in the lower limit topology on ℜ (and sets that are not open) and an explanation of why
• Open sets in the discrete topology on ℜ and an explanation of why there are no sets that are not open
• Two topologies so that the union is not a topology and an explanation of why
• Examples of topologies (without using the indiscrete or discrete topologies) so that the first topology is strictly contained in the second and an explantion of why
• Explanations of why certain topologies are not contained in others (example why isn't the lower limit topology contained in the standard topology?)

How To's

• Given a set of some of the opens, how to generate the rest of the topology
• How to show that something is or is not open in a metric space or a topology
• How to take a union of sets and what the result is
• How to take an intersection of sets and what the result is
• How to complete standard set theory proofs (containment, intersection, union, complement...)

Proofs

• Prove that Bd (x, ε) is an open set.
• Prove that open in ℜ = U (aα, bα)
• Prove that the cofinite topology is coarser than the standard topology on ℜ