FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

TOPOLOGY 9
such that xn ∈ Cn. By hypothesis, {xn} contains a subsequence {xni } which converges to a
point x ∈ A. Choose α such that x ∈ Uα. Since Uα is open in the metric space X, choose
ǫ > 0 so that Bǫ(x) ⊂ Uα, and choose i so large that 1
ni
< ǫ
2 . Then Cni ⊂ Bǫ(x) ⊂ Uα,
which contradicts our supposition.
Lemma. Given ǫ > 0, one can find a finite ǫ-dense subset of A.
Proof. Suppose, on the contrary, that there is an ǫ > 0 such that no finite ǫ-dense set of
X exists. Choose x1 ∈ X. By hypothesis there exists an x2 ∈ X such that x2 /∈ Bǫ(x1). Given
n
x1,...,xn, choose xn+1 such that xn+1 /∈ Bǫ(xi). Since ρ(xn,xj) ≥ ǫ for j = 1,...,n − 1,
i=1
{xn} cannot have a convergent subsequence.
We now conclude the proof of the theorem. Let {Uα} α∈J be an open cover of A. By our
first lemma, we can choose an a > 0 such that for every x ∈ A and δ < a, Bδ(x) ⊂ Uα for
some α ∈ J. Let ǫ = a
3 . By our second lemma, choose a finite ǫ-dense set x1,...,xn in A.
For each 1 ≤ j ≤ n, choose Uj such that Bǫ(xj) ⊂ Uj. Then {Uj} n
j=1
is a finite subcover of
A.
We proceed by developing a bit more machinery involving mappings into metric spaces.
A function f from the space X into the metric space Y is bounded if, for some point y ∈ Y ,
one can find a positive real number M such that f(X) ⊂ BM(y).
If A is a set of bounded functions from the space X into the metric space Y , define the
metric ρ∞(f,g) = supρY
(f(x),g(x)) on A. We call ρ∞ the uniform metric on A. If {fn} is
x
a sequence of functions from X into Y , and {fi(x)} converges for each x ∈ X, then we say
that {fn} converges. Define f : X → Y by f(x) = lim
n→∞ fn(x). We call f the pointwise limit
of {fn}.
Let {fn} be a sequence of functions from X into the metric space Y with pointwise limit
f. If {fn} is fundamental with respect to ρ∞, then {fn} is said to converge to f uniformly.
It is well known that the pointwise limit of a uniformly convergent sequence of continuous
functions is itself continuous.
Let (X,ρX) and (Y,ρY ) be metric spaces. Then f : X → Y is said to be uniformly
continuous if, for every ǫ > 0, one can find a δ > 0 such that for every x ∈ X, x0 ∈ Bδ(x)