nLab
base

Bases and subbases

Idea

For many notions of structure, particularly for topological categories, one can specify a structure by a base or subbase that generate the structure. Besides being convenient ways to specify a structure, they may even be necessary when using weak foundations.

Warning: Sometimes one says ‘basis’ instead of ‘base’, but I think that it's safest to save the former term for the generating set of a free object (or an analogous situation), especially in an algebraic category. Although a basis and a base can both generate something, they tend to do so in very different ways. (It doesn't help that ‘bases’ is the plural of both, although the pronunciation is different.)

Typically, every structure of an appropriate type is both a base and a subbase for itself, while every base is a subbase. Bases and subbases can also be characterised independently; every subbase generates a base (which tends to be saturated in some way), while every base (saturated or not) generates a complete structure.

Examples

Bases for filters

Recall that a filter on a poset LL is a subset FF of LL such that:

  1. If xyx \leq y and xFx \in F, then yFy \in F.
  2. Some xFx \in F.
  3. If x,yFx, y \in F, then for some zFz \in F, zx,yz \leq x, y.

More generally, any subset FF satisfying (2,3) is a filter base.

Given a filter base FF in a poset, we generate a filter F¯\overline{F} by closing under (1); that is, if FF is a filter base on a poset LL, then

F¯{yxF,xy} \overline{F} \coloneqq \{ y \;|\; \exists x \in F,\; x \leq y \}

is a filter on LL.

If LL is a meet-semilattice, then we can equivalently define a filter as a subset FF such that:

  1. If xyx \leq y and xFx \in F, then yFy \in F (same as before).
  2. The top element F\top \in F.
  3. If x,yFx, y \in F, then xyFx \wedge y \in F.

Now any subset satisfying (2,3) is a saturated filter base, and any subset whatsoever is a filter subbase.

Given a filter subbase FF in a semilattice, we generate a base F\vec{F} by closing under (2,3) in the second list; that is, if FF is a filter subbase on a semilattice LL, then

F{ i=1 nx ix iF} \vec{F} \coloneqq \{ \bigwedge_{i=1}^n x_i \;|\; x_i \in F \}

is a filter base on LL, which in fact is saturated. (Note that F\top \in \vec{F} follows when n=0n = 0.)

Given a filter subbase FF in a semilattice, we can generate a filter by first generating a base F\vec{F} and then generating a filter F¯\overline{\vec{F}}. Alternatively, we can generate the same filter by closing under (1,2,3) in the first list all at once. That is, if FF is a filter subbase on a semilattice LL, then

F¯{yx 1,,x nF,zx 1,,x n,zy} \overline{F} \coloneqq \{ y \;|\; \exists x_1,\ldots,x_n \in F,\; \forall z \leq x_1,\ldots,x_n,\; z \leq y \}

is a filter on LL. Furthermore, this is the same filter as F¯\overline{\vec{F}}.

The intersection of any family of filters on a semilattice LL is a filter; that is, being a filter is a Moore closure property on subsets of LL. The filter generated by a filter subbase FF (which is an arbitrary subset of LL, remember) is the same as the Moore closure of FF under this property, that is the intersection of all filters on LL that contain FF.

Unlike filters and filter bases, the concept of filter subbase does not seem to make sense on an arbitrary poset, but only on a semilattice.

Bases for topologies

Recall that a topology on a set XX is a collection 𝒪\mathcal{O} of subsets of XX such that:

  1. Any union of elements of 𝒪\mathcal{O} belongs to 𝒪\mathcal{O}.
  2. XX itself belongs to 𝒪\mathcal{O}.
  3. If U,V𝒪U,V \in \mathcal{O}, then UV𝒪U \cap V \in \mathcal{O}.

More generally, any collection 𝒪\mathcal{O} satisfying (2,3) is a saturated topological base, and any collection whatsoever is a topological subbase.

A slightly more complicated but equivalent definition of topology is this:

  1. Again, any union of elements of 𝒪\mathcal{O} belongs to 𝒪\mathcal{O}.
  2. XX itself is a union of elements of 𝒪\mathcal{O}.
  3. If U,V𝒪U,V \in \mathcal{O}, then UVU \cap V is contained in a union of elements of 𝒪\mathcal{O}.

Now any collection satisfying (2,3) is a topological base (not necessarily saturated).

Given a topological subbase 𝒪\mathcal{O}, we generate a base 𝒪\vec{\mathcal{O}} by closing under (2,3) in the first list; that is, if 𝒪\mathcal{O} is a topological subbase on a set XX, then

𝒪{ i=1 nU iU i𝒪} \vec{\mathcal{O}} \coloneqq \{ \bigcap_{i=1}^n U_i \;|\; U_i \in \mathcal{O} \}

is a topological base on XX, which in fact is saturated. (Note that X𝒪X \in \vec{\mathcal{O}} follows when n=0n = 0.)

Given a topological base 𝒪\mathcal{O}, we generate a topology 𝒪¯\overline{\mathcal{O}} by closing under (1); that is, if 𝒪\mathcal{O} is a topological base on a set XX, then

𝒪¯{VpV,U𝒪,pUV} \overline{\mathcal{O}} \coloneqq \{ V \;|\; \forall p \in V,\; \exists U \in \mathcal{O},\; p \in U \subseteq V \}

is a topology on XX.

Given a topological subbase 𝒪\mathcal{O}, we can generate a topology by first generating a base 𝒪\vec{\mathcal{O}} and then generating a topology 𝒪¯\overline{\vec{\mathcal{O}}}. Alternatively, we can generate the same topology by closing under (1,2,3) in the second list all at once. That is, if 𝒪\mathcal{O} is a topological subbase on a set XX, then

𝒪¯{VpV,U 1,,U n𝒪,p i=1 nU iV} \overline{\mathcal{O}} \coloneqq \{ V \;|\; \forall p \in V,\; \exists U_1, \ldots, U_n \in \mathcal{O},\; p \in \bigcap_{i=1}^n U_i \subseteq V \}

is a topology on 𝒪\mathcal{O}. Furthermore, this is the same topology as 𝒪¯\overline{\vec{\mathcal{O}}}.

As with filters, being a topology is a Moore closure property, this time on subsets of the power set 𝒫X\mathcal{P}X, and the topology generated by a topological subbase 𝒪\mathcal{O} is the intersection of all topologies on XX that contain 𝒪\mathcal{O}.

Very analogous considerations apply to local bases for a topology and bases for pretopologies, convergence structures, gauge structures, Cauchy structures, etc.

Bases for uniformities

Uniformities are a little trickier than topologies, at least in the case of subbases. For now, see the page uniform space for definitions of base and subbase for a uniformity.

Bases for σ\sigma-algebras

Recall that a σ\sigma-algebra on a set XX is

Bases for Grothendieck coverages

See basis for a Grothendieck topology.

General theory

Is there a general theory of bases? That's a good question. I don't know!

Obviously this has something to do with Moore closures (and hence monads); generating a structure from a subbase is (often) taking a Moore closure. But there's some particular property of some closure operators that makes the intermediate concept of base work out.

Revised on March 19, 2014 05:01:34 by Mike Shulman (192.195.154.58)