In

*bounded* open intervals in $\backslash mathbb$ 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 $\backslash mathbb$.
* The set of all bounded *closed* intervals in $\backslash mathbb$ generates 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 $\backslash mathbb$, 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 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 .

^{''n''} is defined by taking the zero sets of polynomial functions as a base for the closed sets.

^{+}.
To see this (without the axiom of choice), fix
$$\backslash left\; \backslash \_,$$
as a basis of open sets. And suppose ''per contra'', that
$$\backslash left\; \backslash \_$$
were a strictly increasing sequence of open sets. This means
$$\backslash forall\; \backslash alpha<\backslash kappa^+:\; \backslash qquad\; V\_\backslash setminus\backslash bigcup\_\; V\_\; \backslash neq\; \backslash varnothing.$$
For
$$x\backslash in\; V\_\backslash setminus\backslash bigcup\_V\_,$$
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
$$V\_\; \backslash setminus\; \backslash bigcup\_\; V\_.$$
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
$$V\_\; \backslash setminus\; \backslash bigcup\_\; V\_\; \backslash subseteq\; V\_\; \backslash setminus\; V\_,$$
which is a contradiction. But this would go to show that ''κ''^{+} ≤ ''κ'', a contradiction.

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 point ...

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. Ideal ...

$\backslash mathcal$ of open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...

s of such that every open set of the topology is equal to the union of some sub-family of $\backslash mathcal$. For example, the set of all open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...

s in the real number line $\backslash R$ 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 $\backslash R$ because every open interval is an open set, and also every open subset of $\backslash R$ 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), edited by Ruth Nanda Anshen
* "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that united the four We ...

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 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 for a topology. Bases for topologies are also closely related to neighborhood bases.
Definition and basic properties

Given atopological 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 point ...

$(X,\backslash tau)$, a baseEngelking, 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 ...

$\backslash tau$ (also called a ''base for'' $X$ 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. Ideal ...

$\backslash mathcal\backslash subseteq\backslash tau$ of open sets such that every open set of the topology can be represented as the union of some subfamily of $\backslash mathcal$.The empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...

, which is always open, is the union of the empty family. The elements of $\backslash mathcal$ are called ''basic open sets''.
Equivalently, a family $\backslash mathcal$ of subsets of $X$ is a base for the topology $\backslash tau$ if and only if $\backslash mathcal\backslash subseteq\backslash tau$ and for every open set $U$ in $X$ and point $x\backslash in\; U$ there is some basic open set $B\backslash in\backslash mathcal$ such that $x\backslash in\; B\backslash subseteq\; U$.
For example, the collection of all open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...

s in the real 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 ...

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 $(X,\backslash tau)$ can have many bases. The whole topology $\backslash tau$ is always a base for itself (that is, $\backslash tau$ is a base for $\backslash tau$). 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
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 $X$ is the minimum cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

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 $\backslash mathcal$ is a base for the topology $\backslash tau$ of a space $X$, it satisfies the following properties:Willard, Theorem 5.3
:(B1) The elements of $\backslash mathcal$ '' cover'' $X$, i.e., every point $x\backslash in\; X$ belongs to some element of $\backslash mathcal$.
:(B2) For every $B\_1,B\_2\backslash in\backslash mathcal$ and every point $x\backslash in\; B\_1\backslash cap\; B\_2$, there exists some $B\_3\backslash in\backslash mathcal$ such that $x\backslash in\; B\_3\backslash subseteq\; B\_1\backslash cap\; B\_2$.
Property (B1) corresponds to the fact that $X$ is an open set; property (B2) corresponds to the fact that $B\_1\backslash cap\; B\_2$ is an open set.
Conversely, suppose $X$ is just a set without any topology and $\backslash mathcal$ is a family of subsets of $X$ satisfying properties (B1) and (B2). Then $\backslash mathcal$ is a base for the topology that it generates. More precisely, let $\backslash tau$ be the family of all subsets of $X$ that are unions of subfamilies of $\backslash mathcal.$ Then $\backslash tau$ is a topology on $X$ and $\backslash mathcal$ is a base for $\backslash tau$.Engelking, Proposition 1.2.1
(Sketch: $\backslash tau$ 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 $\backslash mathcal$. The family $\backslash mathcal$ is then a base for $\backslash tau$ by construction.) Such families of sets are a very common way of defining a topology.
In general, if $X$ is a set and $\backslash mathcal$ is an arbitrary collection of subsets of $X$, there is a (unique) smallest topology $\backslash tau$ on $X$ containing $\backslash mathcal$. (This topology is the intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

of all topologies on $X$ containing $\backslash mathcal$.) The topology $\backslash tau$ is called the topology generated by $\backslash mathcal$, and $\backslash mathcal$ is called a subbase for $\backslash tau$. The topology $\backslash tau$ can also be characterized as the set of all arbitrary unions of finite intersections of elements of $\backslash mathcal$. (See the article about subbase.) Now, if $\backslash mathcal$ also satisfies properties (B1) and (B2), the topology generated by $\backslash mathcal$ can be described in a simpler way without having to take intersections: $\backslash tau$ is the set of all unions of elements of $\backslash mathcal$ (and $\backslash mathcal$ is base for $\backslash tau$ in that case).
There is often an easy way to check condition (B2). If the intersection of any two elements of $\backslash mathcal$ is itself an element of $\backslash mathcal$ or is empty, then condition (B2) is automatically satisfied (by taking $B\_3=B\_1\backslash cap\; B\_2$). 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 $S$ of all semi-infinite intervals of the forms $(-\backslash infty,a)$ and $(a,\backslash infty)$ with $a\backslash in\backslash mathbb$. The topology generated by $S$ contains all open intervals $(a,b)=(-\backslash infty,b)\backslash cap(a,\backslash infty)$, 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 $\backslash mathbb$ forms a basis for theEuclidean 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 $\backslash mathbb$.
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
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), 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 $\backslash mathbb$:
* The set of all 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 t ...

on $\backslash mathbb$ 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 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 ratio ...

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 $\backslash mathbb$.
* generates a topology that is strictly coarser than both the Objects defined in terms of bases

* Theorder 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, t ...

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 setti ...

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 t ...

has the collection of all singletons as a base.
* A second-countable space 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 ...

on the spectrum of a ring
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...

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 ...

of $\backslash C^n$ is the topology that has the algebraic sets 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 t ...

s of algebraic hypersurfaces.
* The Zariski topology of the spectrum of a ring
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...

(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 $\backslash tau\_2$ is finer than a topology $\backslash tau\_1$ if and only if for each $x\backslash in\; X$ and each basic open set $B$ of $\backslash tau\_1$ containing $x$, there is a basic open set of $\backslash tau\_2$ containing $x$ and contained in $B$. * If $\backslash mathcal\_1,\; \backslash ldots,\; \backslash mathcal\_n$ are bases for the topologies $\backslash tau\_1,\; \backslash ldots,\; \backslash tau\_n$ then the collection of all set products $B\_1\; \backslash times\; \backslash cdots\; \backslash times\; B\_n$ with each $B\_i\backslash in\backslash mathcal\_i$ is a base for theproduct 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-seemi ...

$\backslash tau\_1\; \backslash times\; \backslash cdots\; \backslash times\; \backslash tau\_n.$ 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 $\backslash mathcal$ be a base for $X$ and let $Y$ be a subspace of $X$. Then if we intersect each element of $\backslash mathcal$ with $Y$, the resulting collection of sets is a base for the subspace $Y$.
* If a function $f\; :\; X\; \backslash to\; Y$ maps every basic open set of $X$ into an open set of $Y$, it is an open map
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, ...

. Similarly, if every preimage of a basic open set 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 ...

.
* $\backslash mathcal$ is a base for a topological space $X$ if and only if the subcollection of elements of $\backslash mathcal$ which contain $x$ form a local base at $x$, for any point $x\backslash in\; X$.
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 $X,$ afamily
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. Ideal ...

$\backslash mathcal$ 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 $\backslash mathcal$ containing $A$ but not containing $x.$
A family $\backslash mathcal$ is a base for the closed sets of $X$ if and only if its in $X,$ that is the family $\backslash $ of complements of members of $\backslash mathcal$, is a base for the open sets of $X.$
Let $\backslash mathcal$ be a base for the closed sets of $X.$ Then
#$\backslash bigcap\; \backslash mathcal\; =\; \backslash varnothing$
#For each $C\_1,\; C\_2\; \backslash in\; \backslash mathcal$ the union $C\_1\; \backslash cup\; C\_2$ is the intersection of some subfamily of $\backslash mathcal$ (that is, for any $x\; \backslash in\; X$ not in $C\_1\; \backslash text\; C\_2$ there is some $C\_3\; \backslash in\; \backslash mathcal$ containing $C\_1\; \backslash cup\; C\_2$ 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 $\backslash mathcal.$
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
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 ...

on AWeight and character

We shall work with notions established in . Fix ''X'' a topological space. Here, a network is a family $\backslash mathcal$ of sets, for which, for all points ''x'' and open neighbourhoods ''U'' containing ''x'', there exists ''B'' in $\backslash mathcal$ for which $x\; \backslash in\; B\; \backslash subseteq\; U.$ 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, $\backslash chi(x,X),$ as the minimum cardinality of a neighbourhood basis for ''x'' in ''X''; and the character of ''X'' to be $$\backslash chi(X)\backslash triangleq\backslash sup\backslash .$$ 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 $B\text{'}\backslash subseteq\; B$ of size $,\; B\text{'},\; \backslash leq\; w(X).$ * if ''N'' a neighbourhood basis for ''x'' in ''X'' then there is a neighbourhood basis $N\text{'}\backslash subseteq\; N$ of size $,\; N\text{'},\; \backslash leq\; \backslash chi(x,X).$ * if $f\; :\; X\; \backslash to\; Y$ is a continuous surjection, then ''nw''(''Y'') ≤ ''w''(''X''). (Simply consider the ''Y''-network $fB\; \backslash triangleq\; \backslash $ for each basis ''B'' of ''X''.) * if $(X,\backslash tau)$ is Hausdorff, then there exists a weaker Hausdorff topology $(X,\backslash tau\text{'})$ so that $w(X,\backslash tau\text{'})\backslash leq\; nw(X,\backslash tau).$ 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 $f\; :\; X\; \backslash to\; Y$ 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 $nw(f(X))=w(f(X))\backslash leq\; w(X)\backslash leq\backslash aleph\_0$ (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''See also

* Esenin-Volpin's theorem * Gluing axiom * Neighbourhood systemNotes

References

Bibliography

* * * * * * {{refend General topology