Base (topology) explained

l{B}

of open subsets of such that every open set of the topology is equal to the union of some sub-family of

l{B}

. For example, the set of all open intervals in the real number line

\R

is a basis for the Euclidean topology on

\R

because every open interval is an open set, and also every open subset of

\R

can be written as a union of some family of open intervals.

Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called, are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier.

Not all families of subsets of a set

X

form a base for a topology on

X

. Under some conditions detailed below, a family of subsets will form a base for a (unique) topology on

X

, obtained by taking all possible unions of subfamilies. Such families of sets are very frequently used to define topologies. A weaker notion related to bases is that of a subbase for a topology. Bases for topologies are also closely related to neighborhood bases.

Definition and basic properties

(X,\tau)

, a base (or basis) for the topology

\tau

(also called a base for

X

if the topology is understood) is a family

l{B}\subseteq\tau

of open sets such that every open set of the topology can be represented as the union of some subfamily of

l{B}

.[1] The elements of

l{B}

are called basic open sets.Equivalently, a family

l{B}

of subsets of

X

is a base for the topology

\tau

if and only if

l{B}\subseteq\tau

and for every open set

U

in

X

and point

x\inU

there is some basic open set

B\inl{B}

such that

x\inB\subseteqU

.

For example, the collection of all open intervals in the real line forms a base for the standard topology on the real numbers. More generally, in a metric space

M

the collection of all open balls about points of

M

forms a base for the topology.

In general, a topological space

(X,\tau)

can have many bases. The whole topology

\tau

is always a base for itself (that is,

\tau

is a base for

\tau

). For the real line, the collection of all open intervals is a base for the topology. So is the collection of all open intervals with rational endpoints, or the collection of all open intervals with irrational endpoints, for example. Note that two different bases need not have any basic open set in common. One of the topological properties of a space

X

is the minimum cardinality of a base for its topology, called the weight of

X

and denoted

w(X)

. From the examples above, the real line has countable weight.

If

l{B}

is a base for the topology

\tau

of a space

X

, it satisfies the following properties:

(B1) The elements of

l{B}

cover

X

, i.e., every point

x\inX

belongs to some element of

l{B}

.

(B2) For every

B1,B2\inl{B}

and every point

x\inB1\capB2

, there exists some

B3\inl{B}

such that

x\inB3\subseteqB1\capB2

.Property (B1) corresponds to the fact that

X

is an open set; property (B2) corresponds to the fact that

B1\capB2

is an open set.

Conversely, suppose

X

is just a set without any topology and

l{B}

is a family of subsets of

X

satisfying properties (B1) and (B2). Then

l{B}

is a base for the topology that it generates. More precisely, let

\tau

be the family of all subsets of

X

that are unions of subfamilies of

l{B}.

Then

\tau

is a topology on

X

and

l{B}

is a base for

\tau

.(Sketch:

\tau

defines a topology because it is stable under arbitrary unions by construction, it is stable under finite intersections by (B2), it contains

X

by (B1), and it contains the empty set as the union of the empty subfamily of

l{B}

. The family

l{B}

is then a base for

\tau

by construction.) Such families of sets are a very common way of defining a topology.

In general, if

X

is a set and

l{B}

is an arbitrary collection of subsets of

X

, there is a (unique) smallest topology

\tau

on

X

containing

l{B}

. (This topology is the intersection of all topologies on

X

containing

l{B}

.) The topology

\tau

is called the topology generated by

l{B}

, and

l{B}

is called a subbase for

\tau

. The topology

\tau

can also be characterized as the set of all arbitrary unions of finite intersections of elements of

l{B}

. (See the article about subbase.) Now, if

l{B}

also satisfies properties (B1) and (B2), the topology generated by

l{B}

can be described in a simpler way without having to take intersections:

\tau

is the set of all unions of elements of

l{B}

(and

l{B}

is base for

\tau

in that case).

There is often an easy way to check condition (B2). If the intersection of any two elements of

l{B}

is itself an element of

l{B}

or is empty, then condition (B2) is automatically satisfied (by taking

B3=B1\capB2

). For example, the Euclidean topology on the plane admits as a base the set of all open rectangles with horizontal and vertical sides, and a nonempty intersection of two such basic open sets is also a basic open set. But another base for the same topology is the collection of all open disks; and here the full (B2) condition is necessary.

An example of a collection of open sets that is not a base is the set

S

of all semi-infinite intervals of the forms

(-infty,a)

and

(a,infty)

with

a\inR

. The topology generated by

S

contains all open intervals

(a,b)=(-infty,b)\cap(a,infty)

, hence

S

generates the standard topology on the real line. But

S

is only a subbase for the topology, not a base: a finite open interval

(a,b)

does not contain any element of

S

(equivalently, property (B2) does not hold).

Examples

The set of all open intervals in

R

forms a basis for the Euclidean topology on

R

.

A non-empty family of subsets of a set that is closed under finite intersections of two or more sets, which is called a -system on, is necessarily a base for a topology on if and only if it covers . By definition, every σ-algebra, every filter (and so in particular, every neighborhood filter), and every topology is a covering -system and so also a base for a topology. In fact, if is a filter on then is a topology on and is a basis for it. A base for a topology does not have to be closed under finite intersections and many are not. But nevertheless, many topologies are defined by bases that are also closed under finite intersections. For example, each of the following families of subset of is closed under finite intersections and so each forms a basis for some topology on

R

:

R

generates the usual Euclidean topology on

R

.

R

generates the discrete topology on

R

and so the Euclidean topology is a subset of this topology. This is despite the fact that is not a subset of . Consequently, the topology generated by, which is the Euclidean topology on

R

, is coarser than the topology generated by . In fact, it is strictly coarser because contains non-empty compact sets which are never open in the Euclidean topology.

R

.

Objects defined in terms of bases

The Zariski topology on the spectrum of a ring has a base consisting of open sets that have specific useful properties. For the usual base for this topology, every finite intersection of basic open sets is a basic open set.

\Cn

is the topology that has the algebraic sets as closed sets. It has a base formed by the set complements of algebraic hypersurfaces.

Theorems

\tau2

is finer than a topology

\tau1

if and only if for each

x\inX

and each basic open set

B

of

\tau1

containing

x

, there is a basic open set of

\tau2

containing

x

and contained in

B

.

l{B}1,\ldots,l{B}n

are bases for the topologies

\tau1,\ldots,\taun

then the collection of all set products

B1 x x Bn

with each

Bi\inl{B}i

is a base for the product topology

\tau1 x x \taun.

In the case of an infinite product, this still applies, except that all but finitely many of the base elements must be the entire space.

l{B}

be a base for

X

and let

Y

be a subspace of

X

. Then if we intersect each element of

l{B}

with

Y

, the resulting collection of sets is a base for the subspace

Y

.

f:X\toY

maps every basic open set of

X

into an open set of

Y

, it is an open map. Similarly, if every preimage of a basic open set of

Y

is open in

X

, then

f

is continuous.

l{B}

is a base for a topological space

X

if and only if the subcollection of elements of

l{B}

which contain

x

form a local base at

x

, for any point

x\inX

.

Base for the closed sets

Closed sets are equally adept at describing the topology of a space. There is, therefore, a dual notion of a base for the closed sets of a topological space. Given a topological space

X,

a family

l{C}

of closed sets forms a base for the closed sets if and only if for each closed set

A

and each point

x

not in

A

there exists an element of

l{C}

containing

A

but not containing

x.

A family

l{C}

is a base for the closed sets of

X

if and only if its in

X,

that is the family

\{X\setminusC:C\inl{C}\}

of complements of members of

l{C}

, is a base for the open sets of

X.

Let

l{C}

be a base for the closed sets of

X.

Then

capl{C}=\varnothing

  1. For each

C1,C2\inl{C}

the union

C1\cupC2

is the intersection of some subfamily of

l{C}

(that is, for any

x\inX

not in

C1orC2

there is some

C3\inl{C}

containing

C1\cupC2

and not containing

x

).Any collection of subsets of a set

X

satisfying these properties forms a base for the closed sets of a topology on

X.

The closed sets of this topology are precisely the intersections of members of

l{C}.

In some cases it is more convenient to use a base for the closed sets rather than the open ones. For example, a space is completely regular if and only if the zero sets form a base for the closed sets. Given any topological space

X,

the zero sets form the base for the closed sets of some topology on

X.

This topology will be the finest completely regular topology on

X

coarser than the original one. In a similar vein, the Zariski topology on An is defined by taking the zero sets of polynomial functions as a base for the closed sets.

Weight and character

We shall work with notions established in .

Fix X a topological space. Here, a network is a family

l{N}

of sets, for which, for all points x and open neighbourhoods U containing x, there exists B in

l{N}

for which

x\inB\subseteqU.

Note that, unlike a basis, the sets in a network need not be open.

We define the weight, w(X), as the minimum cardinality of a basis; we define the network weight, nw(X), as the minimum cardinality of a network; the character of a point,

\chi(x,X),

as the minimum cardinality of a neighbourhood basis for x in X; and the character of X to be\chi(X)\triangleq\sup\.

The point of computing the character and weight is to be able to tell what sort of bases and local bases can exist. We have the following facts:

B'\subseteqB

of size

|B'|\leqw(X).

N'\subseteqN

of size

|N'|\leq\chi(x,X).

f:X\toY

is a continuous surjection, then nw(Y) ≤ w(X). (Simply consider the Y-network

f'''B\triangleq\{f''U:U\inB\}

for each basis B of X.)

(X,\tau)

is Hausdorff, then there exists a weaker Hausdorff topology

(X,\tau')

so that

w(X,\tau')\leqnw(X,\tau).

So a fortiori, if X is also compact, then such topologies coincide and hence we have, combined with the first fact, nw(X) = w(X).

f:X\toY

a continuous surjective map from a compact metrizable space to an Hausdorff space, then Y is compact metrizable.

The last fact follows from f(X) being compact Hausdorff, and hence

nw(f(X))=w(f(X))\leqw(X)\leq\aleph0

(since compact metrizable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrizable exactly in case they are second countable. (An application of this, for instance, is that every path in a Hausdorff space is compact metrizable.)

Increasing chains of open sets

Using the above notation, suppose that w(X) ≤ κ some infinite cardinal. Then there does not exist a strictly increasing sequence of open sets (equivalently strictly decreasing sequence of closed sets) of length ≥ κ+.

To see this (without the axiom of choice), fix\left \_,as a basis of open sets. And suppose per contra, that\left \_were a strictly increasing sequence of open sets. This means\forall \alpha<\kappa^+: \qquad V_\setminus\bigcup_ V_ \neq \varnothing.

Forx\in V_\setminus\bigcup_V_,we may use the basis to find some Uγ with x in UγVα. In this way we may well-define a map, f : κ+κ mapping each α to the least γ for which UγVα and meetsV_ \setminus \bigcup_ V_.

This map is injective, otherwise there would be α < β with f(α) = f(β) = γ, which would further imply UγVα but also meetsV_ \setminus \bigcup_ V_ \subseteq V_ \setminus V_,which is a contradiction. But this would go to show that κ+κ, a contradiction.

See also

Bibliography

Notes and References

  1. The empty set, which is always open, is the union of the empty family.