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Transfinite Induction and General Topology
Given a set X, the coarsest topology which can be defined on X is the
indiscrete topology in which the only open sets are ∅ and X. At the other
extreme, the finest topology which can be defined on X is the discrete topology
in which all subsets of X are taken as open sets.
A.2.3 Definition If X is a topological space and x ∈ X, then a neighbourhood of x is a set U containing an open set V containing x, that is, x ∈ V ⊆ U ,
where V is open. The neighbourhood system U x of x is the collection of all
neighbourhoods of x.
A.2.4 Definition A neighbourhood base at x in the topological space X is
a subcollection B x ⊆ U x such that, for each U ∈ U x , there exists V ∈ B x
satisfying V ⊆ U . Thus, U x = {U ⊆ X | V ⊆ U for some V ∈ B x }. The
elements of B x are called basic neighbourhoods of x.
A.2.5 Theorem Let X be a topological space, and, for each x ∈ X, let B x
be a neighbourhood base at x. Then the following properties hold.
(a) If V ∈ B x , then x ∈ V .
(b) If V 1 , V 2 ∈ B x , then there is V 3 ∈ B x satisfying V 3 ⊆ V 1 ∩ V 2 .
(c) If V ∈ B x , there is some V 0 ∈ B x such that if y ∈ V 0 , then there is W ∈ B y
satisfying W ⊆ V .
(d) G ⊆ X is open if and only if G contains a basic neighbourhood of each of
its points.
Conversely, suppose that X is a set and that a collection B x of subsets of X,
called basic neighbourhoods of x, is assigned to each element x ∈ X in such a
way that (a), (b), and (c) above are satisfied. If we then define a set G to be
open if and only if it contains a basic neighbourhood of each of its points, as
in (d), we obtain a topology on X in which B x is a neighbourhood base at x
for each x ∈ X.
A.2.6 Definition In a topological space (X, τ ), a base for τ (or a base for X
by an abuse of terminology) is a collection B ⊆ τ of subsets of X such that
each element of τ is a union of elements of B. Equivalently, B is a base for τ if
and only if whenever V ∈ τ and x ∈ V , there is U ∈ B such that x ∈ U ⊆ V .
Furthermore, a collection C ⊆ τ is called a subbase for τ (or a subbase for X)
if the collection of all finite intersections of elements of C forms a base for τ .
A.2.7 Theorem A collection B of subsets of a set X is a base for a topology
on X if and only if the following conditions are satisfied.
(a) B B = X.
∈B
Transfinite Induction and General Topology
Given a set X, the coarsest topology which can be defined on X is the
indiscrete topology in which the only open sets are ∅ and X. At the other
extreme, the finest topology which can be defined on X is the discrete topology
in which all subsets of X are taken as open sets.
A.2.3 Definition If X is a topological space and x ∈ X, then a neighbourhood of x is a set U containing an open set V containing x, that is, x ∈ V ⊆ U ,
where V is open. The neighbourhood system U x of x is the collection of all
neighbourhoods of x.
A.2.4 Definition A neighbourhood base at x in the topological space X is
a subcollection B x ⊆ U x such that, for each U ∈ U x , there exists V ∈ B x
satisfying V ⊆ U . Thus, U x = {U ⊆ X | V ⊆ U for some V ∈ B x }. The
elements of B x are called basic neighbourhoods of x.
A.2.5 Theorem Let X be a topological space, and, for each x ∈ X, let B x
be a neighbourhood base at x. Then the following properties hold.
(a) If V ∈ B x , then x ∈ V .
(b) If V 1 , V 2 ∈ B x , then there is V 3 ∈ B x satisfying V 3 ⊆ V 1 ∩ V 2 .
(c) If V ∈ B x , there is some V 0 ∈ B x such that if y ∈ V 0 , then there is W ∈ B y
satisfying W ⊆ V .
(d) G ⊆ X is open if and only if G contains a basic neighbourhood of each of
its points.
Conversely, suppose that X is a set and that a collection B x of subsets of X,
called basic neighbourhoods of x, is assigned to each element x ∈ X in such a
way that (a), (b), and (c) above are satisfied. If we then define a set G to be
open if and only if it contains a basic neighbourhood of each of its points, as
in (d), we obtain a topology on X in which B x is a neighbourhood base at x
for each x ∈ X.
A.2.6 Definition In a topological space (X, τ ), a base for τ (or a base for X
by an abuse of terminology) is a collection B ⊆ τ of subsets of X such that
each element of τ is a union of elements of B. Equivalently, B is a base for τ if
and only if whenever V ∈ τ and x ∈ V , there is U ∈ B such that x ∈ U ⊆ V .
Furthermore, a collection C ⊆ τ is called a subbase for τ (or a subbase for X)
if the collection of all finite intersections of elements of C forms a base for τ .
A.2.7 Theorem A collection B of subsets of a set X is a base for a topology
on X if and only if the following conditions are satisfied.
(a) B B = X.
∈B