FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

X.
TOPOLOGY 5
Topology
Throughout this section, let X be a set and let ℘(X) denote the class of all subsets of
A topology on X is a subset T of ℘(X) such that:
(1) X and ∅ are elements of T .
n
(2) If U1,...,Un ∈ T , then Ui ∈ T .
i=1
(3) If {Uα} α∈J is a subset of T , then
α∈J
Uα ∈ T .
The set X taken together with the topology T on X is said to be a topological space,
and is sometimes denoted (X, T ). The sets of T are said to be open in X. A subset U of a
topological space X is called Gδ if it can be written as a countable intersection of open sets.
A subset U of X is said to be closed in X if, and only if, X − U is open in X.
Most of the spaces we will study in this paper admit some notion of ‘distance’ between
the elements, and this notion provides critical insights into the structure of the space. We
will, therefore, develop many of our subsequent results with those spaces in mind.
A metric on a set X is a function ρ: X × X → R+ such that for any x,y,z ∈ X,
(1) ρ(x,y) = 0 if, and only if, x = y,
(2) ρ(x,y) = ρ(y,x), and
(3) ρ(x,y) ≤ ρ(x,z) + ρ(z,y).
Metrics are functions which describe the ‘distance’ between any two elements in the space,
and they behave exactly how we think ‘distances’ ought.
If X is a topological space that has a metric ρ defined on it, then for x ∈ X and any
ǫ > 0, we define the ǫ-ball centered at x as Bǫ(x) = {y| ρ(x,y) < ǫ}. We say that the metric
ρ induces the topology of X if Bǫ(x) is open in X for each x ∈ X and ǫ > 0, and if, for any
open set U of X containing the point x ∈ X, there is an ǫ > 0 such that Bǫ(x) ⊂ U. If the
metric ρ induces the topology of X, then (X,ρ) is said to be a metric space.
Let f : X → Y . Then f is said to be continuous if, for each open set U of Y , the pullback
of U under f is open in X. If (X,ρX) and (Y,ρY ) are metric spaces, then this definition