517 HW2
1. Let X be a metric space. A subset E of X is called dense in X if every point of X is a limit
point of E, or a point of E (or both). Prove that the Cantor set is not dense in any interval.
2. A metric space is called separable if it has a countable dense subset. Let X be a metric
space. Prove that X is separable if every infinite subset of X has a limit point.
3. Let X be a metric space. A collection {Uα } of open subsets of X is called a base for X if for
every x ∈ X and every open set V in X containing x, we have x ∈ Uα ⊂ V for some Uα . Prove
that every separable metric space has a countable base.
d
4. Prove that if Un is a dense open subset of Rd for n = 1, 2, . . ., then ∩∞
n=1 Un is dense in R .
P
5. Let X be thePset of sequences x = {xn } of real numbers such that ∞
n=1 |xn | < ∞, and
|x
−
y
|.
Let
z
∈
X
be
a
fixed
sequence
of
positive
real
numbers. Show
define d(x, y) = ∞
n
n=1 n
that E := {x ∈ X : |xn | ≤ zn for all n} is compact.
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