nLab
topological base

Contents

Disambiguation

We discuss here the classical case of bases for topological spaces. For bases on sites, that is for Grothendieck topologies, see Grothendieck pretopology.

Idea

A base or subbase for a topological space is a way of generating its topology from something simpler. This is the appliction to topology of the general concept of base.

Definition

Let XX be a topological space, and let τ\tau be its collection of open subsets (its ‘topology’).

Definition

A base or basis for (or “of”) XX (or τ\tau) is a collection BτB \subset \tau – whose members are called basic open subsets or generating open subsets – such that every open subset is a union of basic ones.

Definition

A subbase for (or “of”) XX (or τ\tau) is a subcollection SτS \subset \tau – whose members are called subbasic open subsets – such that every open subset is a union of finitary intersections of subbasic ones.

If only the underlying set of XX is given, then a base or subbase on this set is any collection of subsets of XX that is a base or subbase for some topology on XX. See below for a characterisation of which collections these can be.

Now fix a point aa in XX.

Definition

A local base or base of neighborhoods or fundamental system of neighborhoods for (or “of”) XX (or τ\tau) at aa is a subcollection BτB \subset \tau – whose members are called basic neighborhoods or generating neighborhoods of aa – such that every basic neighborhood of aa is a neighborhood and every neighborhood of aa is a superset of some basic neighborhood.

We may also allow basic neighborhoods to be non-open, but this really doesn't make any difference; any local base may be refined to a local base of open neighborhoods, and most local bases in practices already come that way.

A local subbase at aa is a family of neighbourhoods aa such that each neighbourhood of aa contains a finite intersection of elements of the family.

Definition

The minimum cardinality of a base of XX is the weight of XX. The minimum cardinality of a base of neighborhoods at aa is the character of XX at aa. The supremum of the characters at all points of XX is the character of XX.

We have assumed the axiom of choice to simplify the description of this concept; but in general one must speak of classes of cardinalities rather than individual cardinalities.

If the character of XX is countable, we say that XX satisfies the first axiom of countability; if the weight is countable, we say that XX satisfies the second axiom of countability.

Examples

Example

For the discrete topology on a set XX, the collection of all singleton subsets is a base, and the singleton {x}\{x\} is a local base at xx. Thus every discrete space is first-countable, but only countable discrete spaces are second-countable.

Example

For every metric space, in particular every paracompact Riemannian manifold, the collection of open subsets that are open balls forms a base for the topology. (For instance, a base for the topology on the real line is given by the collection of open intervals (a,b)(a,b) \subset \mathbb{R}.) Similarly, the collection of open balls containing a given point is a local basis at that point.

Remark

This means that covering families consisting of such basic open subsets are good open covers.

Example

Refining the previous example, every metric space has a basis consisting of the open balls with rational radius. (For instance, a base for the topology on the real line is given by the collection of open intervals (a,b)(a,b) \subset \mathbb{R} where bab - a is rational.) Similarly, the collection of open balls with rational radius containing a given point is a local base at that point. Therefore, every metric space is first-countable.

Example

Now consider a separable metric space; that is, we have a dense subset DD which is countable. Now the space has a basis consisting of the open balls with rational radius and centres in DD. (For instance, a base for the topology on the real line is given by the collection of open intervals (a,b)(a,b) \subset \mathbb{R} where aa and bb are rational.) Therefore, every separable metric space is second-countable.

Generating topologies

Let XX be simply a set.

Proposition

A collection BB of subsets of XX is a base for some topology on XX iff these conditions are met:

  • BB is inhabited;
  • if U,VBU, V \in B, then UVU \cap V contains some element of BB.

This is a sort of colax closure (see discussion) under finitary intersections.

Proposition

Every collection SS of subsets of XX is a subbase for some topology on XX.

A subbase naturally generates a base (for the same topology) by closing it under finitary intersections. (The resulting base will actually be closed under intersection, not just colax-closed.)

Relation to Grothendieck topologies and coverages

If one thinks of the topology on XX as being encoded in the standard Grothendieck topology that it induces on its category of open subsets Op(X)Op(X), then a base for the topology induces a coverage on Op(X)Op(X), whose covering families are the open covers by basic open subsets, which generates this Grothendieck topology.

This coverage is not in general a basis for the Grothendieck topology, because a base for a topological space is in general not closed under intersection with arbitrary open subsets; a coverage is only a basis if is stable under pullback (here, closed under these intersections) and transitive. Unfortunately the established terminology “basis” in topology and topos theory is not quite consistent with the inclusion of topological spaces into topos theory: “basis” in topology corresponds to “coverage” in topos theory, not to “basis” in topos theory.

Revised on August 13, 2014 07:43:26 by Toby Bartels (75.88.46.170)