Math 508
Exam 1
Jerry L. Kazdan
October 12, 2006
12:00 – 1:20
Directions
This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5
points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems
(60 points, 15 points each).
Closed book, no calculators – but you may use one 3
00
×
5
00
card with notes.
Part A: Examples
(4 problems, 5 points each).
Give an example of an inFnite set in a metric
space (perhaps
R
) with the speciFed property.
A–1. Bounded with exactly two limit points.
A–2. Containing all of its limit points.
A–3. Distinct points
{
x
j
}
,
j
= 1
,
2
, . . .
with
x
i
6
=
x
j
for
i
6
=
j
that is compact.
A–4. Closed and bounded but not compact.
Part B: Classify sets
(20 points) ±or each of the following sets,
circle
the listed properties it
has:
a)
{
1 +
1
n
∈
R
, n
= 1
,
2
,
3
, . . .
}
open
closed
bounded
compact
countable
b)
{
1
} ∪ {
1 +
1
n
∈
R
, n
= 1
,
2
,
3
, . . .
}
open
closed
bounded
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