In
mathematics, a base (or basis) for the
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is a
family
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
of
open subsets of such that every open set of the topology is equal to the
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of some
sub-family of
. For example, the set of all
open intervals in the
real number line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
is a basis for the
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot\, ...
on
because every open interval is an open set, and also every open subset of
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
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
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
form a base for a topology on
. Under some conditions detailed below, a family of subsets will form a base for a (unique) topology on
, obtained by taking all possibly 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
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
for a topology. Bases for topologies are also closely related to
neighborhood bases.
Definition and basic properties
Given a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, a base
[Engelking, p. 12] (or basis) for the
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
(also called a ''base for''
if the topology is understood) is a
family
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
of open sets such that every open set of the topology can be represented as the union of some subfamily of
.
[The empty set, which is always open, is the union of the empty family.] The elements of
are called ''basic open sets''.
Equivalently, a family
of subsets of
is a base for the topology
if and only if
and for every open set
in
and point
there is some basic open set
such that
.
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
the collection of all open balls about points of
forms a base for the topology.
In general, a topological space
can have many bases. The whole topology
is always a base for itself (that is,
is a base for
). 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
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
of a space
is the minimum
cardinality of a base for its topology, called the weight of
and denoted
. From the examples above, the real line has countable weight.
If
is a base for the topology
of a space
, it satisfies the following properties:
[Willard, Theorem 5.3]
:(B1) The elements of
''
cover
Cover or covers may refer to:
Packaging
* Another name for a lid
* Cover (philately), generic term for envelope or package
* Album cover, the front of the packaging
* Book cover or magazine cover
** Book design
** Back cover copy, part of co ...
''
, i.e., every point
belongs to some element of
.
:(B2) For every
and every point
, there exists some
such that
.
Property (B1) corresponds to the fact that
is an open set; property (B2) corresponds to the fact that
is an open set.
Conversely, suppose
is just a set without any topology and
is a family of subsets of
satisfying properties (B1) and (B2). Then
is a base for the topology that it generates. More precisely, let
be the family of all subsets of
that are unions of subfamilies of
Then
is a topology on
and
is a base for
.
[Engelking, Proposition 1.2.1]
(Sketch:
defines a topology because it is stable under arbitrary unions by construction, it is stable under finite intersections by (B2), it contains
by (B1), and it contains the empty set as the union of the empty subfamily of
. The family
is then a base for
by construction.) Such families of sets are a very common way of defining a topology.
In general, if
is a set and
is an arbitrary collection of subsets of
, there is a (unique) smallest topology
on
containing
. (This topology is the
intersection of all topologies on
containing
.) The topology
is called the topology generated by
, and
is called a
subbase
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
for
. The topology
can also be characterized as the set of all arbitrary unions of finite intersections of elements of
. (See the article about
subbase
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
.) Now, if
also satisfies properties (B1) and (B2), the topology generated by
can be described in a simpler way without having to take intersections:
is the set of all unions of elements of
(and
is base for
in that case).
There is often an easy way to check condition (B2). If the intersection of any two elements of
is itself an element of
or is empty, then condition (B2) is automatically satisfied (by taking
). For example, the
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot\, ...
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
of all semi-infinite intervals of the forms
and
with
. The topology generated by
contains all open intervals
, hence
generates the standard topology on the real line. But
is only a subbase for the topology, not a base: a finite open interval
does not contain any element of
(equivalently, property (B2) does not hold).
Examples
The set of all open intervals in
forms a basis for the
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot\, ...
on
.
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
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
, every
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
(and so in particular, every
neighborhood filter In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbou ...
), and every
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 aren't. 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
:
* The set of all
bounded open intervals in
generates the usual
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot\, ...
on
.
* The set of all bounded
closed intervals in
generates the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
on
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
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot\, ...
on
, 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.
* The set of all intervals in such that both endpoints of the interval are
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s generates the same topology as . This remains true if each instance of the symbol is replaced by .
* generates a topology that is
strictly coarser than the topology generated by . No element of is open in the Euclidean topology on
.
* generates a topology that is strictly coarser than both the
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot\, ...
and the topology generated by . The sets and are disjoint, but nevertheless is a subset of the topology generated by .
Objects defined in terms of bases
* The
order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, th ...
on a totally ordered set admits a collection of open-interval-like sets as a base.
* In a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
the collection of all
open ball
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defi ...
s forms a base for the topology.
* The
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
has the collection of all
singleton
Singleton may refer to:
Sciences, technology Mathematics
* Singleton (mathematics), a set with exactly one element
* Singleton field, used in conformal field theory Computing
* Singleton pattern, a design pattern that allows only one instance ...
s as a base.
* A
second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
is one that has a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
base.
The
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
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.
* The
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
of
is the topology that has the
algebraic set
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
s as closed sets. It has a base formed by the
set complement
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in .
When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is th ...
s of
algebraic hypersurfaces.
* The Zariski topology of the
spectrum of a ring (the set of the
prime ideals) has a base such that each element consists of all prime ideals that do not contain a given element of the ring.
Theorems
* A topology
is
finer than a topology
if and only if for each
and each basic open set
of
containing
, there is a basic open set of
containing
and contained in
.
* If
are bases for the topologies
then the collection of all
set products with each
is a base for the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-s ...
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.
* Let
be a base for
and let
be a
subspace of
. Then if we intersect each element of
with
, the resulting collection of sets is a base for the subspace
.
* If a function
maps every basic open set of
into an open set of
, it is an
open map. Similarly, if every preimage of a basic open set of
is open in
, then
is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
.
*
is a base for a topological space
if and only if the subcollection of elements of
which contain
form a
local base In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...
at
, for any point
.
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
a
family
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
of closed sets forms a base for the closed sets if and only if for each closed set
and each point
not in
there exists an element of
containing
but not containing
A family
is a base for the closed sets of
if and only if its in
that is the family
of
complements of members of
, is a base for the open sets of
Let
be a base for the closed sets of
Then
#
#For each
the union
is the intersection of some subfamily of
(that is, for any
not in
there is some
containing
and not containing
).
Any collection of subsets of a set
satisfying these properties forms a base for the closed sets of a topology on
The closed sets of this topology are precisely the intersections of members of
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
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
if and only if the
zero set
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or e ...
s form a base for the closed sets. Given any topological space
the zero sets form the base for the closed sets of some topology on
This topology will be the finest completely regular topology on
coarser than the original one. In a similar vein, the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
on A
''n'' 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
of sets, for which, for all points ''x'' and open neighbourhoods ''U'' containing ''x'', there exists ''B'' in
for which
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,
as the minimum cardinality of a neighbourhood basis for ''x'' in ''X''; and the character of ''X'' to be
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:
* ''nw''(''X'') ≤ ''w''(''X'').
* if ''X'' is discrete, then ''w''(''X'') = ''nw''(''X'') = , ''X'', .
* if ''X'' is Hausdorff, then ''nw''(''X'') is finite if and only if ''X'' is finite discrete.
* if ''B'' is a basis of ''X'' then there is a basis
of size
* if ''N'' a neighbourhood basis for ''x'' in ''X'' then there is a neighbourhood basis
of size
* if
is a continuous surjection, then ''nw''(''Y'') ≤ ''w''(''X''). (Simply consider the ''Y''-network
for each basis ''B'' of ''X''.)
* if
is Hausdorff, then there exists a weaker Hausdorff topology
so that
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'').
* if
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
(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
as a basis of open sets. And suppose ''per contra'', that
were a strictly increasing sequence of open sets. This means
For
we may use the basis to find some ''U
γ'' with ''x'' in ''U
γ'' ⊆ ''V
α''. In this way we may well-define a map, ''f'' : ''κ''
+ → ''κ'' mapping each ''α'' to the least ''γ'' for which ''U
γ'' ⊆ ''V
α'' and meets
This map is injective, otherwise there would be ''α'' < ''β'' with ''f''(''α'') = ''f''(''β'') = ''γ'', which would further imply ''U
γ'' ⊆ ''V
α'' but also meets
which is a contradiction. But this would go to show that ''κ''
+ ≤ ''κ'', a contradiction.
See also
*
Esenin-Volpin's theorem
*
Gluing axiom
In mathematics, the gluing axiom is introduced to define what a sheaf (mathematics), sheaf \mathcal F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor
::(X) \rightarrow C
to a cate ...
*
Neighbourhood system In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...
Notes
References
Bibliography
*
*
*
*
*
*
{{refend
General topology