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308 Advanced Linear Algebra
Figure 12.1
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For part 4 , if % cl²:³ , then there are two possibilities. If % : , then the
:
%
constant sequence ²% ³ , with % ~ % for all , is a sequence in that converges
%
:
to . If % ¤ : , then % M ² : ³ and so there is a sequence ² % ³ in for which
%£ % and ²% ³ ¦ %. In either case, there is a sequence in : converging to %.
:
Conversely, if there is a sequence ²% ³ in for which ²% ³ ¦ % , then either
%~ % for some , in which case % : ²:³, or else %£ % for all , in
cl
cl
which case % M²:³ ²:³ .
Dense Subsets
The following concept is meant to convey the idea of a subset : 4 being
“arbitrarily close” to every point in 4 .
Definition A subset of a metric space 4 is in 4 if ² cl : ³dense ~ 4 . A
:
metric space is said to be separable if it contains a countable dense subset.
:
Thus, a subset of 4 is dense if every open ball about any point % 4
contains at least one point of .
:
Certainly, any metric space contains a dense subset, namely, the space itself.
However, as the next examples show, not every metric space contains a
countable dense subset.
Example 12.6
1 The real line is separable, since the rational numbers form a countable
)
r
s
dense subset. Similarly, s is separable, since the set r is countable and
dense.
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2 The complex plane is separable, as is d for all .
d
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3 A discrete metric space is separable if and only if it is countable. We leave
proof of this as an exercise.
