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Unformatted text preview: e that if A and B are sets of real numbers and
ABR
then A B R?
The answer is no. Look at A Q and B R Q.
9. Prove that a subset S of a metric space X is dense in X if and only if we have
SU
whenever U is a nonempty open subset of X.
Suppose that S is dense in X and that U is a nonempty open set. Choose x U. Since x S and U
is a neighborhood of x we know that U S
.
Now suppose that the condition U S
holds for every nonempty open subset U of X. To show
that S X, suppose that x X and that 0. Since the ball B x, is a nonempty open set we must
have B x,
S
.
10. Prove that if U and V are open dense subsets of a metric space X then the set U
if only one of the two sets U and V is open? V is also dense in X. What Solution: All we need to know is that at least one of the sets U and V is open. Suppose that A and B
are sets of real numbers, that
UVX
and that the set U is open.
To prove that
U V X,
suppose that x X and that 0. Since x U we know that the set B x,
U is nonempty and we also
know that this set is open. Therefore, since V X we know that
B x,
UV
.
We have therefore shown that every point of the space X must belong to U V.
11. Skip this exercise if you are not familiar with the concept of a countable set. Find a sequence U n of dense
open subsets of the metric space Q such that
Ý Un
n1 123 . Using the fact that the set Q of rational numbers is countable we express Q in the form
Q r 1 , r 2 , r 3 , , r n , .
We now define
Un rj j n
for each positive integer n. Since the set Q U n is finite for each n we know that each set U n is
open in the metric space Q. Finally, since every neighborhood of a point in Q must contain infiitely
many rational numbers, each neighborhood of a point of Q must intersect with each of the sets
Q U n and so each of the sets Q U n must be dense.
12. Skip this exercise if you are not familiar with the concept of a countable set. Prove that if S is a countable
subset of the metric space R then R S is dense in R. Extend this fact to the metric space R k for an arbitrary
positive integer k.
Suppose that x is a real number and that 0. Since the interval x , x is uncountable we
know that x , x
S
, in other words
x , x
RS
.
Therefore R S is dense in the metric space R.
Now we repeat the same argument in R k . We assume that k is a positive integer and that S is a
countable subset of R k . Suppose that x R k and that 0. Since the ball B x, is uncountable
we know that B x,
S
, in other words
.
B x,
Rk S
Therefore R k S is dense in the metric space R k . 13. Suppose that D is a dense subset of a metric space X and that U is a neighborhood of a point x X. Prove
that there exists a point y D and a rational number r 0 such that
x B y, r
U.
Choose 0 such that B x,
U. Using the fact that D is dense we now choose a member y of
the set D such that d x, y . Finally we choose a positive integer n 3 which makes 1 .
n
3
3 z
y
B y, x
1
n B x, We shall now observe tha...
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 Fall '08
 STAFF
 Math, Calculus

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