B U Department of Mathematics
Fall 2005 Math 331  Real Analysis I,
Final Exam, 5/1/2006, 15:0017:30
Whatever theorem, proposition, lemma, exercise,
example you use, you must EXACTLY STATE it
Full Name
:
(its hypothesis and its conclusion). Besides, either
it MUST have been already proven in the class or
Student ID
:
you must PROVE it now. If you do not state what
you are using or do not prove, your solutions will
not be accepted.
I encourage you make use of
theorems.
Do not forget to return this exam sheet!
over 60
Below,
C
(
X, Y
)
and
C
b
(
X, Y
)
denote the set of all continuous functions (and bounded in the latter) from
X
to
Y
equipped with the sup metric whenever it makes sense.
The answers to questions 13 are to be written on this sheet just below the question.
1. (1) What do we mean when we say ”pointwise” and ”uniform”?
2. (2) Sketch a closed subset
C
of
R
2
such that
π
(
C
)
⊂
R
is
not
closed, where
π
:
R
2
→
R
is the
projection given by
π
(
x, y
) =
x
(which is a continuous function).
3. (3) Let
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 Spring '11
 talinbudak
 Math, Topology, Metric space, Compact space, closed unit disk

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