Base (topology)

A sub-base in a topological space is a set of open sets which clearly describes the topology. Is this system ( as defined below ) also closed with respect to the intersection form, then one speaks of a base of the topological space. A local variant is the term neighborhood basis, see also environment.

Topological spaces which have countable neighborhood bases or bases that satisfy the first and second countability axiom. You can in the topological sense are considered "small ".

  • 4.1 Local definition of continuity
  • 4.2 Initial topology

Definitions

Basis of a topology

  • A system B of subsets of a topological space is called a basis of the topology, and topological basis if
  • Conversely, one can a system B of subsets of a set X with the properties

Sub-base of a topology

  • A quantity system of subsets of a ground set

Ie sub-base of the topological space if

Holds, where the closure operator designates which produces the minimum over a system topology of subsets of. For then applies

The closure operator provides a topology, since the average over any number of topologies is a topology again.

  • Conversely, any system of subsets of a set of defining a topology ( defined by the topology ) can be used to:

Neighborhood basis

For one is called the system of neighborhoods of x neighborhood basis of x with respect to the topological space if every open neighborhood of x contains an environment as a subset.

Properties

  • Is a sub-base, then the set of all finite intersections of sets in a topological basis.
  • Each topological basis of a sub-base of, the basic concept thus exacerbated the term sub-base.
  • The notion of topological basis is not to be confused with the basis of a vector space, the former is a lot of open sets, temptation a set of vectors in the case of topological vector spaces so a lot of points. The terms indicate the extent a parallel in that both produce the overall structure in a certain sense, however, is required minimality for a topological basis in any way.
  • Usually one is actually interested in the smallest possible bases, but there are generally no useful formalized description for a "minimal" topological basis. For example, the set of all open sets of a topological basis every topological space.
  • Is a neighborhood basis given for each point x of X of a topological space, then the union of all these surrounding bases forms a base of the topological space.
  • A worsening of these: If D is a dense subset of the space X, then the system is already around bases to the elements of D is a topological basis of the space.

Examples

  • The set B = { X} generated as a topological basis, the indiscrete topology on X in which only the empty set and X is open.
  • For the quantity system is the topology generated:
  • The system of one-point sets { x} of a space X is defined as a topological basis, the discrete topology on X in which all subsets of X are open.
  • If { x} is open, then { { x} } neighborhood basis of x.
  • For a natural number, the quantity system of open intervals is on a base of the natural topology.
  • In a metric space the amount of ε - neighborhoods of a point x is a basis of neighborhoods of x, the set of all ε - balls a base of the topology induced by the metric.

Applications

Local definition of continuity

The concept of neighborhood basis allows a convenient characterization of continuity: Are and topological spaces and f is a mapping from X to Y, then f is continuous at the point x of X if for environment bases in X and in Y:

For the special case where X and Y are metric spaces and are selected as environment bases, the ε - environment, this is the " ε - δ definition of consistency ," which is preferably in the elemental analysis with respect to the general topological definition.

Initial topology

The concept of sub-base allows a constructive definition of initial topologies: The initial topology for an illustration of the sub-base is just the inverse images of open sets in generated topology.

Trivial topology
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