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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a base or basis for the
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
is a
family In human society A society is a Social group, group of individuals involved in persistent Social relation, social interaction, or a large social group sharing the same spatial or social territory, typically subject to the same Politic ...
of
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
s of such that every open set is equal to a
union
union
of some
sub-family
sub-family
of (this sub-family is allowed to be infinite, finite, or even emptyBy a standard convention, the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

empty set
, which is always open, is the union of the empty collection.
). For example, the set of all
open interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s in the
real number line Real may refer to: Currencies * Brazilian real The Brazilian real ( pt, real, plural, pl. '; currency symbol, sign: R$; ISO 4217, code: BRL) is the official currency of Brazil. It is subdivided into 100 centavos. The Central Bank of Brazil ...

real number line
\R is a basis for the
Euclidean topologyIn mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition In any metric space, the Ball (mathematics), ope ...
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 Convergence may refer to: Arts and media Literature *Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-par ...
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 form a base for a topology. For example, because is always an open subset of every topology on , if a family of subsets is to be a base for a topology on then it must
cover Cover or covers may refer to: Packaging, science and technology * A covering, usually - but not necessarily - one that completely closes the object ** Cover (philately), generic term for envelope or package ** Housing (engineering), an exterior ...
, which by definition means that the union of all sets in must be equal to . If has more than one point then there exist families of subsets of that do not cover and consequently, they can not form a basis for topology on . A family of subsets of that does form a basis for some topology on is called a , in which case this necessarily unique topology, call it , is said to be and is consequently a basis for topology . Such families of sets are frequently used to define topologies. A weaker notion related to bases is that of a subbasis for a topology. Bases for topologies are closely related to neighborhood bases.


Definition and basic properties

A base is a collection ''B'' of open subsets of ''X'' satisfying the following properties: # The base elements ''
cover Cover or covers may refer to: Packaging, science and technology * A covering, usually - but not necessarily - one that completely closes the object ** Cover (philately), generic term for envelope or package ** Housing (engineering), an exterior ...
'' ''X''. # Let ''B''1, ''B''2 be base elements and let ''I'' be their intersection. Then for each ''x'' in ''I'', there is a base element ''B''3 containing ''x'' such that ''B''3 is subset of ''I''. An equivalent property is: any finite intersectionWe are using a convention that the empty intersection of subsets of ''X'' is considered finite and is equal to ''X''. of elements of ''B'' can be written as a union of elements of ''B''. These two conditions are exactly what is needed to ensure that the set of all unions of subsets of ''B'' is a topology on ''X''. If a collection ''B'' of subsets of ''X'' fails to satisfy these properties, then it is not a base for ''any'' topology on ''X''. (It is a
subbase In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...
, however, as is any collection of subsets of ''X''.) Conversely, if ''B'' satisfies these properties, then there is a unique topology on ''X'' for which ''B'' is a base; it is called the topology generated by ''B''. (This topology is the
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
of all topologies on ''X'' containing ''B''.) This is a very common way of defining topologies. A sufficient but not necessary condition for ''B'' to generate a topology on ''X'' is that ''B'' is closed under intersections; then we can always take ''B''3 = ''I'' above. For example, the collection of all
open interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s in the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
forms a base for a topology on the real line because the intersection of any two open intervals is itself an open interval or empty. In fact they are a base for the standard topology on the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. However, a base is not unique. Many different bases, even of different sizes, may generate the same topology. For example, the open intervals with rational endpoints are also a base for the standard real topology, as are the open intervals with irrational endpoints, but these two sets are completely disjoint and both properly contained in the base of all open intervals. In contrast to a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
of a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
, a base need not be maximal; indeed, the only maximal base is the topology itself. In fact, any open set generated by a base may be safely added to the base without changing the topology. The smallest possible
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of a base is called the weight of the topological space. An example of a collection of open sets which is not a base is the set ''S'' of all semi-infinite intervals of the forms (−∞, ''a'') and (''a'', ∞), where ''a'' is a real number. Then ''S'' is ''not'' a base for any topology on R. To show this, suppose it were. Then, for example, (−∞, 1) and (0, ∞) would be in the topology generated by ''S'', being unions of a single base element, and so their intersection (0,1) would be as well. But (0, 1) clearly cannot be written as a union of elements of ''S''. Using the alternate definition, the second property fails, since no base element can "fit" inside this intersection. Given a base for a topology, in order to prove convergence of a net or sequence it is sufficient to prove that it is eventually in every set in the base which contains the putative limit.


Examples

The set of all open intervals in form a basis for the
Euclidean topologyIn mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition In any metric space, the Ball (mathematics), ope ...
on . Every topology on a set is a basis for itself (that is, is a basis for ). Because of this, if a theorem's hypotheses assumes that a topology has some basis , then this theorem can be applied using . 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 Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ** Filter (aquarium), critical ...
(and so in particular, every neighborhood filter), and every
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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 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 topologyIn mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition In any metric space, the Ball (mathematics), ope ...
on . * The set of all bounded closed intervals in generates the
discrete topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
on and so the Euclidean topology is a subset of this topology. This is despite the fact that is not a subset . Consequently, the topology generated by , which is the
Euclidean topologyIn mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition In any metric space, the Ball (mathematics), ope ...
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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
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 topologyIn mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition In any metric space, the Ball (mathematics), ope ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is usually defined as the topology generated by a collection of open-interval-like sets. * The
metric topology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is usually defined as the topology generated by a collection of
open ball In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. * A
second-countable space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), a ...
is one that has a
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
base. * The
discrete topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
has the singletons as a base. * The profinite topology on a group is defined by taking the collection of all normal subgroups of finite index as a basis of open neighborhoods of the identity. The
Zariski topology In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commut ...
on the
spectrum of a ring In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
has a base consisting of open sets that have specific useful properties. For the usual basis of this topology, every finite intersection of basis elements is a basis element. Therefore bases are sometimes required to be stable by finite intersection. * The
Zariski topology In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commut ...
of \mathbb C^n is the topology that has the
algebraic set Algebraic may refer to any subject related to algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geomet ...
s as closed sets. It has a basis formed by the
set complement In , the complement of a , often denoted by (or ), are the not in . When all sets under consideration are considered to be s of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of ...
s of algebraic hypersurfaces. * The Zariski topology of the
spectrum of a ring In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
(the set of the prime ideals) has a basis such that each element consists of all prime ideals that do not contain a given element of the ring.


Theorems

* For each point ''x'' in an open set ''U'', there is a base element containing ''x'' and contained in ''U''. * A topology ''T''2 is finer than a topology ''T''1
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
for each ''x'' and each base element ''B'' of ''T''1 containing ''x'', there is a base element of ''T''2 containing ''x'' and contained in ''B''. * If ''B''1,''B''2,...,''B''''n'' are bases for the topologies ''T''1,''T''2,...,''T''''n'', then the set product ''B''1 × ''B''2 × ... × ''B''''n'' is a base for the
product topology Product may refer to: Business * Product (business) In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer ...
''T''1 × ''T''2 × ... × ''T''''n''. 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 ''B'' be a base for ''X'' and let ''Y'' be a subspace of ''X''. Then if we intersect each element of ''B'' with ''Y'', the resulting collection of sets is a base for the subspace ''Y''. * If a function ''f'' : ''X'' → ''Y'' maps every base element of ''X'' into an open set of ''Y'', it is an
open map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. Similarly, if every preimage of a base element of ''Y'' is open in ''X'', then ''f'' 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 ga ...
. * A collection of subsets of ''X'' is a topology on ''X'' if and only if it generates itself. * ''B'' is a basis for a topological space ''X'' if and only if the subcollection of elements of ''B'' which contain ''x'' form a
local baseIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
at ''x'', for any point ''x'' of ''X''.


Base for the closed sets

Closed set In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
s 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 of closed sets ''F'' 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 ''F'' containing ''A'' but not containing ''x''. It is easy to check that ''F'' is a base for the closed sets of ''X'' if and only if the family of complements of members of ''F'' is a base for the open sets of ''X''. Let ''F'' be a base for the closed sets of ''X''. Then #''F'' = ∅ #For each ''F''1 and ''F''2 in ''F'' the union ''F''1 ∪ ''F''2 is the intersection of some subfamily of ''F'' (i.e. for any ''x'' not in ''F''1 or ''F''2 there is an ''F''3 in ''F'' containing ''F''1 ∪ ''F''2 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 ''F''. 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. Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, w ...
if and only if the
zero set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
s 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 In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commut ...
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 \mathcal of sets, for which, for all points ''x'' and open neighbourhoods ''U'' containing ''x'', there exists ''B'' in \mathcal for which ''x'' ∈ ''B'' ⊆ ''U''. 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: * ''nw''(''X'') ≤ ''w''(''X''). * if ''X'' is discrete, then ''w''(''X'') = ''nw''(''X'') = , ''X'', . * if ''X'' is Hausdorff, then ''nw''(''X'') is finite iff ''X'' is finite discrete. * if ''B'' is a basis of ''X'' then there is a basis B'\subseteq B of size , B', \leq w(X). * if ''N'' a neighbourhood basis for ''x'' in ''X'' then there is a neighbourhood basis N'\subseteq N of size , N', \leq \chi(x,X). * if ''f'' : ''X'' → ''Y'' is a continuous surjection, then ''nw''(''Y'') ≤ ''w''(''X''). (Simply consider the ''Y''-network fB \triangleq \ for each basis ''B'' of ''X''.) * if (X,\tau) is Hausdorff, then there exists a weaker Hausdorff topology (X,\tau') so that w(X,\tau')\leq nw(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''). * if ''f'' : ''X'' → ''Y'' a continuous surjective map from a compact metrisable space to an Hausdorff space, then ''Y'' is compact metrisable. The last fact follows from ''f''(''X'') being compact Hausdorff, and hence nw(f(X))=w(f(X))\leq w(X)\leq\aleph_0 (since compact metrisable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrisable exactly in case they are second countable. (An application of this, for instance, is that every path in an Hausdorff space is compact metrisable.)


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. For :x\in V_\setminus\bigcup_V_, we may use the basis to find some ''Uγ'' with ''x'' in ''Uγ'' ⊆ ''Vα''. In this way we may well-define a map, ''f'' : ''κ''+ → ''κ'' mapping each ''α'' to the least ''γ'' for which ''Uγ'' ⊆ ''Vα'' and meets :V_ \setminus \bigcup_ V_. This map is injective, otherwise there would be ''α'' < ''β'' with ''f''(''α'') = ''f''(''β'') = ''γ'', which would further imply ''Uγ'' ⊆ ''Vα'' but also meets :V_ \setminus \bigcup_ V_ \subseteq V_ \setminus V_, 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
* Neighbourhood system *
Subbase In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...


Notes


References


Bibliography

* * * * * * * * * *
James Munkres James Raymond Munkres (born August 18, 1930) is a Professor Emeritus of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (ge ...
(1975) ''Topology: a First Course''. Prentice-Hall. * * * * * * Willard, Stephen (1970) ''General Topology''. Addison-Wesley. Reprinted 2004, Dover Publications. {{DEFAULTSORT:Base (Topology) General topology