Wikipedia / Base (topology)
Wikipedia / Base (topology)
In mathematics, a base (or basis; : bases) for the topology of a topological space is a family of open subsets such that every open set of the topology is equal to the union of some sub-family. For example, the set of all open intervals in the real number line is a basis for the Euclidean topology can be written as a union of some family of open intervals.
Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called , are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier.
, obtained by taking all possible unions of subfamilies. Such families of sets are very frequently used to define topologies. A weaker notion related to bases is that of a subbase for a topology. Bases for topologies are also closely related to neighborhood bases.
Definition and basic properties
Given a topological space, a base (or basis) for the topology (also called a base for if the topology is understood) is a family. are called basic open sets.
For example, the collection of all open intervals in the real line forms a base for the topology.
). 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 is the minimum cardinality of a base for its topology, called the weight. From the examples above, the real line has countable weight.
, it satisfies the following properties:
.
is an open set.
. by construction.) Such families of sets are a very common way of defining a topology.
. (This topology is the intersection is called the topology generated by is called a subbase (see the article about subbase in that case).
). For example, the Euclidean topology on the plane admits as a base the set of all open rectangles with horizontal and vertical sides, and a nonempty intersection of two such basic open sets is also a basic open set. But another base for the same topology is the collection of all open disks; and here the full (B2) condition is necessary.
(equivalently, property (B2) does not hold).
Examples
forms a basis for the Euclidean topology.
that is closed under finite intersections of two or more sets, which is called a \pi-system. By definition, every σ-algebra, every filter (and so in particular, every neighborhood filter), and every topology is closed under finite intersections and so each forms a basis for some:
of all bounded generates the usual Euclidean topology.
of all bounded closed generates the discrete topology, which is the Euclidean topology, is coarser than. In fact, it is strictly contains non-empty compact sets which are never open in the Euclidean topology.
such that both endpoints of the interval are rational numbers.
generates a topology that is strictly coarser.
generates a topology that is strictly coarser than both the Euclidean topology.
Objects defined in terms of bases
The order topology on a totally ordered set admits a collection of open-interval-like sets as a base.
In a metric space the collection of all open balls forms a base for the topology.
The discrete topology has the collection of all singletonss as a base.
A second-countable space is one that has a countable base.
The Zariski topology on the spectrum of a ring has a base consisting of open sets that have specific useful properties. For the usual base for this topology, every finite intersection of basic open sets is a basic open set.
The Zariski topology is the topology that has the algebraic sets as closed sets. It has a base formed by the set complements of algebraic hypersurfacess.
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
is finer.
then the collection of all set products is a base for the product topology 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.
be a subspace.
, it is an open map is continuous.
form a local base.
Base for the closed sets
Closed sets a family of closed sets forms a base for the closed sets if and only if its of complements
Then
).
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 coarser than the original one. In a similar vein, the Zariski topology 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 .
a topological space. Here, a network and open neighbourhoods U Note that, unlike a basis, the sets in a network need not be open.
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; and the character
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:
.
.
is finite discrete.
.)
So a fortiori.
a continuous surjective map from a compact metrizable space to an Hausdorff space is compact metrizable.
(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
.
as a basis of open sets. And suppose per contra
, a contradiction.