Basic Analysis – Gently Done Topological Vector Spaces

2 Basic Analysis
Definition 1.3 A topological space (X,T) is said to be metrizable if there is a
metric on X such that T is as in example 5 above.
Remark 1.4 Noteverytopologyismetrizable. Thespaceinexample2aboveisnot
metrizable whenever X consists of more than one point. (If it were metrizable and
contained distinct points a and b, say, then the set {x : d(x,b) < d(a,b)} would
be a non-empty proper open subset.) Of course, even if a topological space is
metrizable, the metric will be far from unique—for example, proportional metrics
generate the same collection of open sets.
Definition 1.5 For any non-empty subset A of a topological space (X,T), the
induced (or relative)topology, T A , onAisdefined tobethat givenby thecollection
A∩T = {A∩U : U ∈ T} of subsets of A. (It is readily verified that T A is a topology
on A.)
Many of the usual concepts in metric space theory also appear in that of
topological spaces—but suitably rephrased in terms of open sets.
Definition 1.6 Suppose that (X,T) is a topological space. A subset F of X is said
to be closed if and only if its complement X \ F is open, that is, belongs to T.
(It follows immediately that if {F α } is any collection of closed sets then �
α F α is
closed. Indeed, X \ �
α F α
= �
α (X \F α
), which belongs to T.)
A point a ∈ X is an interior point of the subset A of X if there is U ∈ T such
that a ∈ U and U ⊆ A. (Thus, a set G is open if and only if each of its points
is an interior point of G. To see this, suppose that each point of G is an interior
point of G. Then for each x ∈ G there is U x ∈ T such that x ∈ U x ⊆ G. Hence
G = �
x∈G U x
∈ T. The converse is clear—take U = G.)
The set of interior points of the set A is denoted by ◦
A, or IntA.
The point x is a limit point (or accumulation point) of the set A if and only
if for every open set U containing x, it is true that U ∩ A contains some point
distinct from x, i.e., A∩{U \{x}} �= ∅. Note that x need not belong to A.
The point a ∈ A is said to be an isolated point of A if there is an open set U
such that U ∩A = {a}. (In other words, there is some open set containing a but
no other points of A.)
The closure of the set A, written A, is the union of A and its set of limit points,
A = A∪{x ∈ X : x is a limit point of A}.
It follows from the definition that x ∈ A if and only if A ∩ U �= ∅ for any
open set U containing x. Indeed, suppose that x ∈ A and that U is some open
set containing x. Then either x ∈ A or x is a limit point of A (or both), in which
case A∩U �= ∅. On the other hand, suppose that A∩U �= ∅ for any open set U
containing x. Then if x is not an element of A it is certainly a limit point. Thus
x ∈ A.
Department of Mathematics King’s College, London