Base (topology)
Encyclopedia
In mathematics
, a base (or basis) B for a topological space
X with topology
T is a collection of open set
s in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases are useful because many properties of topologies can be reduced to statements about a base generating that topology, and because many topologies are most easily defined in terms of a base which generates them.
If a collection B of subsets of X fails to satisfy either of these, then it is not a base for any topology on X. (It is a subbase
, however, as is any collection of subsets of X.) Conversely, if B satisfies both of the conditions 1 and 2, 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
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 B3 = I above.
For example, the collection of all open intervals in the real line
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
s.
However, a base is not unique. Many 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
of a vector space
in linear algebra
, a base need not be maximal
; indeed, the only maximal base is the topology itself. In fact, any open sets in the space generated by a base may be safely added to the base without changing the topology. The smallest possible cardinality 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 the 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.
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 base for the closed sets of X is a family of closed sets F such that any closed set A is an intersection
of members of F.
Equivalently, a family of closed sets forms a base for the closed sets 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
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 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 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.
a topological Space. We define the Weight as 
the minimum Cardinality of a Basis; we define the network Weight as 
the minimum Cardinality of a Network; the Character of a Point 
the minimum Cardinality of a Neighbourhood Basis for 
in 
; and the Character of 
to be 
.
Here, a Network is a family
of sets, for which, for all Points 
and open Neighbourhoods 
, there is a 
for which 
.
The point of computing the Character and Weight is useful to be able to tell what sort of Bases and local Bases can exist. We have following Facts:
The last Fact comes from the Fact that
is compact hausdorff, and hence 
(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.)
some infinite Cardinal.
Then there does not exist a strictly increasing Sequence of open Sets (equivalently
strictly decreasing Sequence of closed Sets) of Length
.
To see this (without the Axiom of Choice), fix
a Basis of open Sets. And suppose per contra, that 
were a strictly increasing Sequence of open Sets. This means 
is non-empty. If 
, we may utilise the Basis to find some 
with 
. In this way we may well-define a Map, 
mapping each 
to the least 
for which 
and meets
. This Map can be seen to be injective. (For otherwise there would be 
with 
, say, which would further imply 
but also meets 
which is a Contradiction.) But this would go to show that 
, a Contradiction.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a base (or basis) B for a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X with topology
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
T is a collection of open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases are useful because many properties of topologies can be reduced to statements about a base generating that topology, and because many topologies are most easily defined in terms of a base which generates them.
Simple properties of bases
Two important properties of bases are:- The base elements coverCover (topology)In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset. Formally, ifC = \lbrace U_\alpha: \alpha \in A\rbrace...
X. - Let B1, B2 be base elements and let I be their intersection. Then for each x in I, there is a base element B3 containing x and contained in I.
If a collection B of subsets of X fails to satisfy either of these, then it is not a base for any topology on X. (It is a subbase
Subbase
In topology, a subbase for a topological space X with topology T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B...
, however, as is any collection of subsets of X.) Conversely, if B satisfies both of the conditions 1 and 2, 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
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
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 B3 = I above.
For example, the collection of all open intervals in the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
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
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s.
However, a base is not unique. Many 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 (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
of a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
in linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, a base need not be maximal
Maximal
Maximal may refer to:*Maximal element, a mathematical definition*Maximal , a faction of Transformers*Maximalism, an artistic style*Maximal set*Maxim , a men's magazine marketed as Maximal in several countriesSee also...
; indeed, the only maximal base is the topology itself. In fact, any open sets in the space generated by a base may be safely added to the base without changing the topology. The smallest possible cardinality 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 the 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.
Objects defined in terms of bases
- The order topologyOrder topologyIn 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...
is usually defined as the topology generated by a collection of open-interval-like sets. - The metric topology is usually defined as the topology generated by a collection of open balls.
- A second-countable spaceSecond-countable spaceIn topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
is one that has a countable base. - The discrete topology has the singletons as a base.
Theorems
- For each point x in an open set U, there is a base element containing x and contained in U.
- A topology T2 is finerTopological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
than a topology T1 if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
for each x and each base element B of T1 containing x, there is a base element of T2 containing x and contained in B. - If B1,B2,...,Bn are bases for the topologies T1,T2,...,Tn, then the set productCartesian productIn mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
B1 × B2 × ... × Bn is a base for the product topologyProduct topologyIn 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...
T1 × T2 × ... × Tn. 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 subspaceTopological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
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. Similarly, if every preimage of a base element of Y is open in X, then f is continuous.
- 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 base at x, for any point x of X.
Base for the closed sets
Closed setClosed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
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 base for the closed sets of X is a family of closed sets F such that any closed set A is an intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of members of F.
Equivalently, a family of closed sets forms a base for the closed sets 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
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
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 F1 and F2 in F the union F1 ∪ F2 is the intersection of some subfamily of F (i.e. for any x not in F1 or F2 there is an F3 in F containing F1 ∪ F2 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 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 finest completely regular topology on X coarser than the original one. In a similar vein, the Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
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 a topological Space. We define the Weight as the minimum Cardinality of a Basis; we define the network Weight as the minimum Cardinality of a Network; the Character of a Point the minimum Cardinality of a Neighbourhood Basis for in ; and the Character of to be .Here, a Network is a family
of sets, for which, for all Points and open Neighbourhoods , there is a for which .The point of computing the Character and Weight is useful to be able to tell what sort of Bases and local Bases can exist. We have following Facts:
- obviously
. - if
is discrete, then 
. - if
is hausdorff, then 
is finite iff 
is finite discrete. - if
a Basis of 
then there is a Basis 
of Size 
. - if
a Neighbourhood Basis for 
then there is a Neighbourhood Basis 
of Size 
. - if
is a continuous surjection, then 
. (Simply consider the 
-Network 
for each Basis 
of 
.) - if
is hausdorff, then there exists a weaker hausdorff Topology 
so that 
. So a forteori, if 
is also compact, then such Topologies coincide and hence we have, combined with the first Fact, 
. - if
a continuous surjective map from a compact metrisable Space to an hausdorff Space, then 
is compact metrisable.
The last Fact comes from the Fact that
is compact hausdorff, and hence (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 given Notation, suppose that some infinite Cardinal.Then there does not exist a strictly increasing Sequence of open Sets (equivalently
strictly decreasing Sequence of closed Sets) of Length
.To see this (without the Axiom of Choice), fix
a Basis of open Sets. And suppose per contra, that were a strictly increasing Sequence of open Sets. This means is non-empty. If , we may utilise the Basis to find some with . In this way we may well-define a Map, mapping each to the least for which and meets
. This Map can be seen to be injective. (For otherwise there would be with , say, which would further imply but also meets which is a Contradiction.) But this would go to show that , a Contradiction.