nLab topological base

Contents

This page discusses bases for the topology on topological spaces. For the concept of topological linear basis see at basis in functional analysis. For bases on sites, that is for Grothendieck topologies, see at Grothendieck pretopology.

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

A base and a subbase for a topological space are ways of generating its topology from something simpler. This is the application 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.

Properties

Generating topologies

Let XX be simply a set.

Proposition

(recognition of topological bases)

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

  • The elements of BB cover XX;
  • For any U,VBU, V \in B and any point xUVx \in U \cap V there is a WBW \in B such that WUVW \subseteq U \cap V and xWx \in W.

These conditions amount to saying that for each xXx\in X, the subcollection of those UBU\in B such that xUx\in U is a base for a filter on XX (which is then the neighborhood filter of xx) — in other words, that these subcollections are “colaxly closed” under finite 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.)

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.

Literature

  • Ryszard Engelking, General Topology, PWN; Warszawa 1977; Revised and completed edition in: Sigma series in pure mathematics 6, Heldermann 1989 ISBN 388538-006-4

  • Parimal, Proofs (?), Lecture 13: Basis for a topology

Last revised on October 10, 2023 at 11:52:28. See the history of this page for a list of all contributions to it.