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 ℜ^{2} 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 B_{d} (x, ε) is an open set.
Prove that open in ℜ = U (a_{α}, b_{α})
Prove that the cofinite topology is coarser than the standard topology
on ℜ