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

<p><b>Definitions</b>
<li>Set containment
<li>Function
<li>Inverse image of a function
<li>Intersection
<li>Union
<li>Complement
<li>Continuity in &real;
<li>Metric space
<li>Continuity in a metric space
<li>Metric ball
<li>Open set in a metric space
<li>Topology
<li>Open in a topology
<li>Comparing topologies

<p><b>Examples</b>
<li>Discrete topology and the metric it arises from
<li>Standard topology on &real; and a metric it arises from
<li>Square metric
<li>Taxicab metric
<li>Standard topology on &real;<sup>2</sup> and two different metrics it arises 
from
<li>Sierpinski space
<li>Cofinite topology
<li>Lower limit topology
<li>Open sets in the standard toplogy on &real; (and sets that are not 
open) and an explanation of why
<li>Open sets in the cofinite topology on &real; (and sets that are not open) and an explanation of why 
<li>Open sets in the lower limit topology on &real; 
(and sets that are not open) and an explanation of why 
<li>Open sets in the discrete topology on &real; and an explanation of why
there are no sets that are not open
<li>Two topologies so that the union is not a topology and an explanation 
of why 
<li>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 
<li>Explanations of why certain topologies are not contained in others 
(example why isn't 
the lower limit topology contained in the standard topology?) 

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


<p><b>Proofs</b>
<li>Prove that B<sub>d</sub> (x, &epsilon;) is an open set. 
<li>Prove that open in &real; = U (a<sub>&alpha;</sub>, b<sub>&alpha;</sub>)
<li>Prove that the cofinite topology is coarser than the standard topology
on &real;