FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

TOPOLOGY 7
Proof. (Necessity) Let x ∈ X, and {xn} be a sequence converging to x. Given ǫ > 0,
choose δ > 0 such that if y ∈ Bδ(x), then f(y) ∈ Bǫ (f(x)). Choose an N ∈ N such that for
every n > N, xn ∈ Bδ(x). Then f(xn) ∈ Bǫ (f(x)).
(Sufficiency) Suppose that f is not continuous at a point x ∈ X. Let δn = 1
. Then for
n
some ǫ > 0, choose for each n ∈ N a xn such that xn ∈ Bδn(x), but f(xn) /∈ Bǫ (f(x)). Then
{xn} converges to x, but {f(xn)} does not converge to f(x), contrary to hypothesis.
If A is a subset of the metric space Y , the the set of points {yα} α∈J is said to be ǫ-dense
with respect to A if, for each a in A, one can find an α ∈ J such that ρ(a,yα) < ǫ.
If U is a subset of a topological space X, then we say that x ∈ X is a limit point of U
if every open set of X containing x also contains a point y of U. If X is a metric space,
then this definition is equivalent to saying that x is a limit point of U if, and only if, there
is a sequence {un} in U which converges to x in X. The closure A of a subset A of the
topological space X is the set A together with the limit points of A. A can also be described
as the smallest closed set containing A.
A subset U of a topological space X is dense in X if U = X. The following theorem will
prove useful in our later discussion.
in V .
Theorem. U is dense in X if, and only if, for every open set V of X, U ∩ V is dense
Proof. Suppose U is dense in X, and that V is an open subset of X, Then U ∩ V ⊂
U ∩ V = X ∩ V = V . If x ∈ V , and if O is an open set containing x, then there is a point
v ∈ O such that v ∈ V . Since O ∩ V is open, and v ∈ O ∩ V , then the density of U in X
guarantees that there is a point u ∈ O ∩ V ⊂ O, such that u ∈ U. Hence, V ⊂ U ∩ V , so
that U ∩ V = V . The converse follows immediately by noting that X is an open subset of
itself.
One of the most important and useful properties that a topological space can possess is
called compactness. Let A be a subset of the topological space X, and {Uα} α∈J be a class of
open sets in X. If A ⊂
α∈J
Uα, we say that {Uα} α∈J is an open covering of A. If J ∗ ⊂ J, and