User:Mgkrupa/Filter bases in topology

In functional analysis and related areas of mathematics, a topological vector spaces (TVS) is complete if its canonical uniformity is complete.

A Metrizable topological vector space X with a translation invariant metric d is complete as a TVS if and only if (X, d) is a complete metric space. All topological vector spaces, even those that are not metrizable or Haussddorff, have a completion.

Definitions and notation

Main article: Filter (mathematics)

Throughout, X will be a non-empty set and 𝒜 and ℬ will be collections of subsets of X.

The theory of filters and filter bases is well developed and has many definitions and notations, many of which we now unceremoniously list to prevent this article from becoming prolix and to allow for the easy look up of notation and definitions. We describe many of their important properties later. Note that not all notation related to filters is well established and some notation varies greatly across the literature (e.g. the notation for the set of all filter bases on a set) so in such cases this article uses whatever notation is most self describing or easily remembered.

Notation	Definition	Name
℘(X)  :=  { S : S ⊆ X }	Set of all subsets of X	Power set of a set X
Top(X)	Set of all topologies on X
Filters(X)	Set of all filters on X
PreFilters(X) = FilterBases(X)	Set of all filter bases on X
UltraFilters(X)	Set of all ultrafilters on X
Func(X; Y)	Set of all functions from X into Y

Sets operations

Definition: The upward closure or isotonization of a collection ℬ of subsets of X is

ℬ^↑  :=  ℬ^↑X  :=  { S ⊆ X : B ⊆ S for some B ∈ ℬ }  =  ∪B ∈ ℬ { S : B ⊆ S ⊆ X }.

Notation and Definition	Assumptions	Name
ker ℬ  :=  ∩B ∈ ℬ B		Kernel
ℬ ∩ { S }  :=  { B ∩ S : B ∈ ℬ }	S ⊆ X	Trace of ℬ on S
S ∖ ℬ  :=  { S ∖ B : B ∈ ℬ }	S ⊆ X	Set subtraction
ℬ↓  :=  ℬ↓X  =  ∪B ∈ ℬ { S : S ⊆ B }	S ⊆ X	Downward closure
𝒜 + ℬ  :=  { A + B : A ∈ 𝒜, B ∈ ℬ }	X is an additive group	Sum
s ℬ  :=  { s B : B ∈ ℬ }	X is a vector space and s is a scalar	Scalar multiple
f (ℬ)  :=  { f (B) : B ∈ ℬ }	f : X → Y is a map	Image of ℬ under f
f -1 (𝒞)  :=  { f -1 (C) : C ∈ 𝒞 }	f : X → Y is a map and 𝒞 ⊆ ℘(Y)	Preimage of ℬ under f
Im f  :=  f (X)  =  { f (x) : x ∈ X }	f : X → Y is a map	Image or range of f
S↑X  :=  { S }↑X	S ⊆ X	Upward closure/Isotonization
𝒜 ⊓ ℬ  :=  𝒜 ∩ ℬ  :=  { A ∩ B : A ∈ 𝒜, B ∈ ℬ }		Pairwise intersection

Nets and topologies

Directed sets and nets notation.

Definition: A directed set is a set I together with a preorder, which we will assume is denoted by ≤ (unless otherwise specified), that makes (I, ≤) into an upward directed set, which means that for every i, j ∈ I, there exists some k ∈ I such that i ≤ k and j ≤ k. We define j ≥ i to mean i ≤ j. A net in X is a map from a directed set into X.

Notation and Definition	Assumptions	Name
I≥ i  :=  { j ∈ I : i ≤ j }	i ∈ I and (I, ≤) is a directed set	Tail of I starting at i
I> i  :=  { j ∈ I : i ≤ j,  j ≠ i }	i ∈ I and (I, ≤) is a directed set	Tail of I after i
f (I≥ i)  =  { f (j) : i ≤ j,  j ∈ I }	i ∈ I and f : (I, ≤) → X is a net	Tail of f starting at i
f (I> i)  =  { f (j) : i ≤ j,  j ≠ i,  j ∈ I }	i ∈ I and f : (I, ≤) → X is a net	Tail of f after i
(x•)i  :=  x≥ i  :=  { xj : i ≤ j,  j ∈ I }	i ∈ I and x• = (xi)i ∈ I is a net	Tail of x• starting at i
x> i  :=  { xj : i ≤ j,  j ≠ i,  j ∈ I }	i ∈ I and x• = (xi)i ∈ I is a net	Tail of x• after i
Tails(x•)  :=  (xi) i  :=  x≥ •  :=  { x≥ i : i ∈ I }	x• = (xi)i ∈ I is a net	Set/(Eventuality) filter base of/associated with/generated by (tails of) x•. If x• is a sequence it is called the sequential filter base instead.
TailsFilter(x•)  :=  Tails(x•)↑X	x• = (xi)i ∈ I is a net	(Eventuality) filter of/associated with/generated by (tails of) x•

Topology notation

If τ is a topology on X then we may use the following notation.

Notation and Definition	Assumptions	Name
τ (S)  :=  { O ∈ τ : S ⊆ O }	S ⊆ X	(Filter base of) Open neighborhoods of S in (X, τ)
τ (x)  :=  { O ∈ τ : x ∈ O }	x ∈ X	(Filter base of) Open neighborhoods of x in (X, τ)
𝒩τ(S)  :=  𝒩(S)  :=  τ (S)↑X	S ⊆ X	(Filter of) Neighborhoods of S in (X, τ)
𝒩τ(x)  := 𝒩(x)  :=  τ (x)↑X	x ∈ X	(Filter of) Neighborhoods of x in (X, τ)
𝒩τ : X → Filter(X)    is the map    x ↦ 𝒩τ(x)		Map of neighborhood filters (induced by τ) from X

Finer, coarser, subordinate, product

The following definition of 𝒜 ≤ ℬ allows for the filter equivalent of "subsequence." It will also be used to define convergence. The definition of 𝒜 meshes with ℬ is used in Topology to define cluster points.

1. 𝒜 ≤ ℬ or ℬ ⊢ 𝒜, stated as ℬ is finer than 𝒜, 𝒜 is coarser than ℬ, or ℬ is subordinate to 𝒜: If any of the following equivalent conditions hold:

   1. Every A ∈ 𝒜 contains some B ∈ ℬ.
   2. For every A ∈ 𝒜, there is some B ∈ ℬ such that B ⊆ A.
   3. 𝒜 ⊆ ℬ^↑X.

   Thus if ℬ is upward closed (i.e. isotone) then we may add to this list:

   4. 𝒜 ⊆ ℬ.
      • So in this case, this definition of "ℬ is finer than 𝒜" would be identical to the topological definition of "finer" had 𝒜 and ℬ been topologies on X.

   • Note that if 𝒜 ⊆ ℬ then 𝒜 ≤ ℬ. Thus ℬ ≤ ℘(X) is always true, for any ℬ ⊆ ℘(X).
   • ≤ is transitive: 𝒜 ≤ ℬ and ℬ ≤ 𝒞 implies 𝒜 ≤ 𝒞.
   • ≤ is reflexive: 𝒜 ≤ 𝒜 is always true.

2. 𝒜 < ℬ, stated as ℬ is strictly finer than 𝒜: If 𝒜 ≤ ℬ and 𝒜 ≠ ℬ.

3. 𝒜 is equivalent to ℬ: If any of the following equivalent conditions hold:

   1. 𝒜 ≤ ℬ and ℬ ≤ 𝒜.
   2. The upward closures of 𝒜 and ℬ are equal.

   • The relation ≤ is not antisymmetric; that is, 𝒜 ≤ ℬ and ℬ ≤ 𝒜 does not necessarily imply ℬ = 𝒜; not even if both 𝒜 and ℬ are filter bases. However, ≤ is transitive and reflexive so this definition does indeed define an equivalence relation.

4. 𝒜 ◅ ℬ or ℬ ▻ 𝒜 stated as 𝒜 is a refinement of ℬ, 𝒜 refines ℬ: If any of the following equivalent conditions hold:

   1. Every A ∈ 𝒜 is contained in some B ∈ ℬ.
   2. For every A ∈ 𝒜, there is some B ∈ ℬ such that A ⊆ B.
   3. (X ∖ 𝒜) ≤ (X ∖ ℬ).

5. 𝒜 # ℬ, stated as 𝒜 and ℬ mesh or are compatible: if A ∩ B ≠ ∅ for all A ∈ 𝒜 and B ∈ ℬ.

   • If S ⊆ X then we write S # ℬ to mean { S } # ℬ.
   • If 𝒜 and ℬ are filter bases on X then 𝒜 and ℬ mesh if and only if there exists a filter base 𝒞 on X such that 𝒜 ≤ 𝒞 and ℬ ≤ 𝒞, or equivalently, if and only if 𝒜 ∩ ℬ is a filter base on X.

6. 𝒜 and ℬ are dissociated: if 𝒜 and ℬ do not mesh.

7. Product: Suppose X_• = (X_i)_{i ∈ I} is a non-empty collection of non-empty sets and that ℬ_• = (ℬ_i)_{i ∈ I} is a collection of non-empty sets where each ℬ_i ⊆ ℘(X_i). We define the product of the sets ℬ_• in the same way that one would define the product topology had all of these ℬ_i been topologies. That is,

        ∏ ℬ_• := ∏i ∈ I ℬ_i

   denotes the collection of all subsets ∏i ∈ I S_i of ∏ X_• := ∏i ∈ I X_i such that S_i = X_i for all but finitely many i ∈ I and for any one of these finitely many i that satisfy S_i ≠ X_i, we necessarily have S_i ∈ ℬ_i.

Filters and filter bases

We now define properties that a collection ℬ ⊆ ℘(X) may have.

Definitions: We say that ℬ is/has the:

1. Proper: ∅ ∉ ℬ.
2. Degenerate: ∅ ∈ ℬ.
3. Closed under (finite) intersections: If B, C ∈ ℬ then B ∩ C ∈ ℬ.
4. Directed by (superset/reverse) inclusion: If B, C ∈ ℬ then there exists some A ∈ ℬ such that A ⊆ B ∩ C.
   • Equivalently, ℬ is a directed set when we define B ≤ A if and only if A ⊆ B for all subsets A and B of X.
5. Finite intersection property and is centralized: The intersection of any finite collection of sets in ℬ is not empty. That is, if B₁, ..., B_n ∈ ℬ then ∅ ≠ B₁ ∩ ⋅⋅⋅ ∩ B_n.
6. Upward closed/Isotone: If ℬ = ℬ^↑X, or equivalently, if whenever B ∈ ℬ and C is a set such that B ⊆ C ⊆ X then C ∈ ℬ.
   • ℬ^↑X is the a unique smallest isotone collection of subsets of X, in which case we may say that ℬ^↑X is generated by ℬ.
7. Ultra: For any S ⊆ X there exists some B ∈ ℬ such that B ⊆ S or B ⊆ X ∖ S (or equivalently, such that B ∩ S equals B or ∅).

The definitions above are all that is needed to define filters, prefilters, filer subbases, and ultrafilters, which we now define.

Definitions: If ℬ ⊆ ℘(X) then we say that ℬ is/is a(n):

1. Dual ideal: If ℬ ≠ ∅ is closed under (finite) intersections and upward closed.
2. Filter: if ℬ ≠ ∅ is proper, closed under (finite) intersections, and upward closed. Equivalently, a filter is a proper dual ideal.
   • The intersection of any non-empty set 𝔽 of filters on X is a filter on X, called the infimum or greatest lower bound of 𝔽. In contrast, the least upper bound of a family of filters may fail to be a filter.
3. Subfilter of a filter ℱ: if ℬ is a filter and ℬ ≤ ℱ (where since ℱ is isotone, ℬ ≤ ℱ if and only if ℬ ⊆ ℱ).
4. Filter base/Prefilter: if ℬ ≠ ∅ is proper and directed by superset inclusion. Equivalently, it is a subset of ℘(X) whose upward closure forms a filter on X.
   • If ℬ is a filter base then the upward closure ℬ^↑X is the unique filter containing ℬ called the filter generated by ℬ. We say that a filter ℱ is generated by a filter base ℬ if ℱ = ℬ^↑X, in which case we say that ℬ is a filter base for ℱ.
5. Filter subbase: if ℬ ≠ ∅ has the finite intersection property (which implies that ℬ is proper).
   • The collection of all finite intersections of subsets of a filter subbase ℬ is a filter base called the filter base generated by ℬ and the filter generated by this filter base is called the filter generated by the filter subbase ℬ.
   • A collection of sets ℬ ≠ ∅ is contained in a filter if and only if it has the finite intersection property.
6. Maximal/Ultra prefilter: If ℬ is a filter base and ℬ satisfies any of the following equivalent conditions:
   1. For any filter base 𝒟 on X, if ℬ ≤ 𝒟 then 𝒟 ≤ ℬ.
   2. ℬ is ultra;
   • It may be shown using Zorn's lemma that for every filter base, there is a maximal filter base subordinate to it.
7. Ultrafilter: If ℬ is a filter and it satisfies any of the following equivalent conditions:
   1. ℬ is ultra;
   2. For any subset S ⊆ X, S ∈ ℬ or X ∖ S ∈ ℬ.
      • So an ultrafilter ℬ decides for every S ⊆ X whether S is "large" (i.e. S ∈ ℬ) or "small" (i.e. X ∖ S ∈ ℬ).
   3. ℬ ∪ (X ∖ ℬ) = ℘(X) (where ℬ ∩ (X ∖ ℬ) = ∅ is true for any filter base ℬ).
   4. For any finite collection S₁, ..., S_n of subsets of X, if S₁ ∪ ⋅⋅⋅ ∪ S_n ∈ ℬ then S_i ∈ ℬ for some index i.
      • Thus a "large" set cannot be a finite union of small sets.
   5. For any subsets R, S ⊆ X, if R ∪ S ∈ ℬ then R ∈ ℬ or S ∈ ℬ (a filter with this property is called a prime filter).
   6. For any subsets R, S ⊆ X such that R ∩ S = ∅, if R ∪ S ∈ ℬ then either R ∈ ℬ or S ∈ ℬ.
   7. ℬ is a maximal filter; that is, if ℱ is a filter on X such that ℬ ⊆ ℱ then ℬ = ℱ. Equivalently, ℬ is a maximal filter if there is no filter ℱ on X that contains ℬ as a proper subset.
   • If ℬ ≠ ∅ is any collection of subsets of X, then ℬ is an ultrafilter on X if and only if ℬ has the finite intersection property and for any S ⊆ X, either S ∈ ℬ or X ∖ S ∈ ℬ.

We now list some more properties that ℬ may have.

Definitions: We say that ℬ is:

8. The Fréchet filter: if X is infinite and ℬ is equal to the set of all cofinite subsets of X (sets whose complement in X is finite).
9. Free: ker ℬ = ∅.
   • A prefilter is free if and only if ℬ contains the Fréchet filter on X.
10. Fixed: ker ℬ ≠ ∅ in which case we say that ℬ is fixed by any point x ∈ ker ℬ.
11. Principal: ker ℬ ∈ ℬ.
12. Discrete/Principal at x ∈ X: { x } = ker ℬ ∈ ℬ.
    • The principal filter at x is the filter { x }^↑X. A filter ℱ is principal at x if and only if ℱ = { x }^↑X.
13. Indiscrete: ℬ = { X }.
14. Additive: For every B ∈ ℬ, there exists some U ∈ ℬ such that U + U ⊆ B (this assumes that X is a group).
    • If ℬ is a filter then this happens if and only if ℬ ⊆ ℬ + ℬ.

Examples and properties

Images and preimages of filter bases

• If ℬ is a prefilter (resp. ultra prefilter) on X then f (ℬ) is a prefilter (resp. ultra prefilter) on Y.
• If ℬ is a filter on X then f (ℬ)^↑Y := { S ⊆ Y : f ^-1 (S) ∈ ℬ }, and f (ℬ) is a filter on Y if and only if f is surjective.
• If ℬ is a prefilter on X then f ^-1 (f (ℬ)) is a prefilter on X and moreover, f ^-1 (f (ℬ)) ≤ ℬ.
• If ℬ is an ultrafilter on X and f is surjective then f (ℬ) is an ultrafilter on Y.
• A map f : X → Y is injective if and only if whenever ℱ is a filter on Y then f ^-1 (ℱ ) is a filter on X.
• If f is a bijection then ℬ is a prefilter (resp. filter, ultrafilter) on X if and only if the same is true of f (ℬ) on Y.
• If 𝒞 is a filter base on Y then f ^-1 (𝒞) is a filter base on X if and only if ∅ ∉ f ^-1 (𝒞), in which case 𝒞 ≤ f (f ^-1 (𝒞)).
• If 𝒞 is a filter on Y then f ^-1 (𝒞) is a filter base on X but it may fail to be a filter on X even if f is surjective.

Properties of ultrafilters

• A free ultrafilter on X exists if and only if X is infinite.
  • A filter ℱ on a finite set X is an ultrafilter if and only if it is principal at some x ∈ X (i.e. if and only if ker ℱ = { x } ∈ ℱ for some x ∈ X).
• An ultra prefilter generates an ultrafilter.
• Every filter is equal to the intersection of all ultrafilters containing it.
• If ∅≠ ℬ ⊆ ℘(X) then ℬ can be extended to a free ultrafilter if and only if the intersection of any finite collection of elements of ℬ is infinite.
• A principal filter 𝒫 on X (i.e. a filter that satisfies ker 𝒫 ∈ 𝒫) is an ultrafilter if and only if ker 𝒫 is a singleton set. All other ultrafilters on 𝒰 are free (i.e. ker 𝒰 = ∅).

The ultrafilter theorem/principle/lemma — Every filter (and thus also prefilter and filter subbase) on a set X is contained in some ultrafilter on X.

Proposition — If 𝒰 is an ultrafilter on a non-empty set X then the following are equivalent:

1. 𝒰 is not free.
2. 𝒰 is fixed (i.e. ker 𝒰 ≠ ∅).
3. 𝒰 is principle.
4. 𝒰 is principle at some x ∈ X (i.e. ker 𝒰 = { x } ∈ 𝒰 for some x ∈ X).
5. ker 𝒰 is a singleton set contained in 𝒰.
6. 𝒰 contains a finite subset of X.
7. 𝒰 does not contain the Fréchet filter on X.

Examples of filter bases and filters

• The intersection of any non-empty family of filters on X is a filter on X; moreover, it is the largest filter contained in each member of this family.
  • However, if X contains at least 2 distinct elements then there exist filters ℬ and 𝒞 on X for which there does not exist a filter ℱ on X that contains both ℬ and 𝒞.
• If ℬ is a filter base then the trace of ℬ on S is a filter base if B ∩ S ≠ ∅ for all B ∈ ℬ.

Properties

• If 𝒜 ≤ ℬ and ℬ is a filter base then 𝒜 and ℬ mesh (see footnote for proof).

Filter bases on topological spaces

Throughout, (X, τ) is a topological space.

A note on intuition

Suppose that ℱ is a non-principal filter on an infinite set X. ℱ has one "upward" property (that of being closed upward) and one "downward" property (that of being directed downwards under set inclusion). Starting with any F₀ ∈ ℱ, there always exists some F₁ ∈ ℱ that is a proper subset of F₀; this may be continued ad infinitum to get F₀ ⊃ F₁ ⊃ F₂ ⊃ ⋅⋅⋅ in ℱ with each F_{i + 1} being a proper subset of F_i. The same is not true going "upward", for if F₀ = X ∈ ℱ then there is no set in ℱ that contains X as a proper subset. Thus when it comes to limiting behavior (which is a topic central to the field of topology), going "upward" leads to a dead end, while going "downward" is typically fruitful. So to gain understanding and intuition about how filters (and filter bases) relate to topology, the "downward" property is usually the one to concentrate on. This is also why so many topological properties can be described by using only filter bases, rather than requiring filters (which only differ from filter bases in that they are also upward closed).

Topological definitions

Definition: If S ⊆ X and x ∈ X then we say that x is a limit point, cluster point, or accumulation point of S if every neighborhood of x in (X, τ) contains a point of S different from x, or equivalently, if x ∈ Cl_{(X, τ)} (S ∖ { x  }). The set of all limit points of S is called the derived set of S in (X, τ).

Recall that the closure of a set S ⊆ X is equal to the union of S together with the set of all limit points of S.

Limit and cluster points of filters

Definition: We say a collection ℬ ⊆ ℘(X) converges to a point x ∈ X in (X, τ), that x is a limit point of ℬ, and we write ℬ → x in X if 𝒩(x) ≤ ℬ (i.e. if ℬ is finer than 𝒩(x)). Explicitly, this means that every neighborhood N of x contains some element of ℬ as a subset. Notation: lim ℬ will denote the set of all limit points of ℬ in (X, τ).

Definition: If S ⊆ X and ℬ ⊆ ℘(X) then we say that ℬ converges to S in (X, τ), that S is a limit of ℬ, and we write ℬ → S in X if 𝒩(S) ≤ ℬ.

In the above definitions, it suffices to check that ℬ is finer than some (or equivalently, all) neighborhood base in (X, τ) of x or S .

Definition: If x ∈ X and ℬ ⊆ ℘(X) then we call x a cluster point or accumulation point of ℬ if ℬ meshes with the neighborhood filter at x; that is, if B ∩ N ≠ ∅ for every B ∈ ℬ and every neighborhood N of x in X.

If ℬ is a prefilter on X then the set of all cluster points of ℬ is equal to ∩B ∈ ℬ cl B, which justifies the following notation.

Notation: We denote the set of all cluster points of ℬ by cl ℬ.

Definition: If S ⊆ X and ℬ ⊆ ℘(X) then we say that ℬ clusters at S if ℬ meshes with the neighborhood filter of S; that is, if B ∩ N ≠ ∅ for every B ∈ ℬ and every neighborhood N of S in X.

In the above definitions, it suffices to check that ℬ meshes with some (or equivalently, all) neighborhood base in (X, τ) of x or S. Clearly, ℬ converges to (resp. clusters at) x if and only if ℬ converges to (resp. clusters at) { x } if and only if the filter ℬ^↑X generated by ℬ converges to (resp. clusters at) x. If x is a limit point of ℬ then x is a limit point of any any family 𝒞 finer than ℬ (i.e. if 𝒩(x) ≤ ℬ and ℬ ≤ 𝒞 then 𝒩(x) ≤ 𝒞). In contrast, if x is a cluster point of ℬ then x is a cluster point of any family 𝒜 coarser than ℬ (i.e. if 𝒩(x) and ℬ mesh and 𝒜 ≤ ℬ then 𝒩(x) and 𝒜 mesh).

Every limit point of a filter base ℬ is a cluster point of ℬ, since if x is a limit point of a filter base ℬ then 𝒩(x) and ℬ mesh and thus x is a cluster point of ℬ. If 𝒰 is an ultra prefilter on X and x ∈ X, then x is a cluster point of 𝒰 if and only if 𝒰 → x (X, τ). The set of all cluster points of a filter base cl ℬ in a topological space X is a closed subset of X and moreover, cl ℬ = ∩ { cl B : B ∈ ℬ}.

A point x ∈ X is a cluster point of a prefilter ℬ if and only if there exists a finer filter base 𝒞 (i.e. ℬ ≤ 𝒞) such that x is a limit point of 𝒞.

Filters and nets

The relation ≤ is of fundamental importance to applying filters to topology. We may use the ≤ relation to define the analogue of "subsequence" for filter bases and also to define convergence for filter bases. We will use these definitions to characterize in terms of filters and filter bases concepts like continuity and limits of functions.

Nets as prefilters

We now define limits and cluster points of nets. In the definitions below, we start with the standard definition of a limit point of a net (resp. a cluster point of a net) and gradually reword it until we arrive at the corresponding filter concept.

Definition: We say that an X-valued net x_• = (x_i)_{i ∈ I} converges in (X, τ) to a point x ∈ X, written x_• → x in (X, τ), and call x a limit point of x_• if any of the following equivalent conditions hold:

1. For every neighborhood N of x in (X, τ), there exists some i ∈ I such that if i ≤ j ∈ I then x_j ∈ N.
2. For every neighborhood N of x in (X, τ), there exists some i ∈ I such that the tail of x_• starting at i is contained in N.
3. For every neighborhood N of x in (X, τ), there exists some B ∈ Tails(x_•) such that B ⊆ N.
4. The filter base Tails(x_•) converges to x in (X, τ); that is Tails(x_•) → x in (X, τ).

Definition: We say that x ∈ X is a cluster point or accumulation point of an X-valued net x_• = (x_i)_{i ∈ I} in (X, τ) if any of the following equivalent conditions hold:

1. For every neighborhood N of x in (X, τ) and every i ∈ I, there exists some i ≤ j ∈ I such that x_j ∈ N.
2. For every neighborhood N of x in (X, τ) and every i ∈ I, the tail of x_• starting at i intersects N ("intersects" means that the intersection is not empty).
3. For every neighborhood N of x in (X, τ) and every B ∈ Tails(x_•), B ∩ N ≠ ∅.
   • Thus the neighborhood filter of x in (X, τ) and the filter base Tails(x_•) mesh (by definition of "mesh").
4. x is a cluster point of Tails(x_•) in (X, τ).

• The notion of "ℬ is subordinate to 𝒜" (written ℬ ⊢ 𝒜) is for filters and filter bases what "x_n• = (x_ni)^∞
  _i=1 is a subsequence of x_• = (x_i)^∞
  _i=1" is for sequences (and nets).
  • Indeed, if we let 𝒳 := { x_{≥ i} : i ∈ ℕ  } denotes the set of tails of x_• and 𝒮 denotes the set of tails of the subsequence x_n•, then 𝒮 ⊢ 𝒳 (i.e. 𝒳 ≤ 𝒮) is true but 𝒳 ⊢ 𝒮 is in general false.
• If x_• = (x_i)_{i ∈ I} is a net in a topological space X, 𝒳 := { x_{≥ i} : i ∈ I  } is the set of its tails, and 𝒩(x) is the neighborhood filter at a point x ∈ X, then x_• → x in X if and only if 𝒩(x) ≤ 𝒳.

Definition: A net x_• in X is called an ultranet or universal net in X if for every subset S ⊆ X, x_• is eventually in S or it is eventually in X ∖ S.

A net x_• = (x_i)_{i ∈ I} in X is an ultranet if and only if Tails(x_•) is an ultra prefilter.

Prefilters as nets

Definition: Recall that a pointed set is a pair (S, s) consisting of a non-empty set S and an element s ∈ S. If ℬ is a collection of non-empty sets then let

     PointedSets( ℬ )

denoted the set of all pointed sets (B, b) such that B ∈ ℬ and b ∈ B. If (B, b), (C, c) ∈ PointedSets( ℬ ) then we declare that

     (B, b) ≤ (C, c) if and only if C ⊆ B.

Observe that if ℬ is a prefilter on X then PointedSets( ℬ ) is a directed set so if we desire a map of the form PointedSets( ℬ ) → X then the obvious choice is the assignment (B, b) ↦ b.

Definition: If ℬ is a prefilter on X then let the net associated with ℬ be the map      Net_ℬ : PointedSets( ℬ ) → X      defined by      Net_ℬ(B, b) := b.

One may show that if ℬ is a prefilter on X then Net_ℬ : PointedSets( ℬ ) → X is a net in X, and that the prefilter associated with Net_ℬ is Tails(Net_ℬ) = ℬ. This would not necessarily be true had Net_ℬ been defined on a proper subset of PointedSets( ℬ ). For instance, if X has at least two distinct elements, ℬ := { X  } is the indiscrete filter on X, x ∈ X is arbitrary, and Net_ℬ been defined on the singleton set D := { (X, x) }, then the prefilter associated with Net_ℬ : D → X would be the principal prefilter { x } rather than ℬ = { X  } (where note that ℬ is the unique minimal filter on X whereas { x } generates a maximal/ultrafilter on X).

However, if x_• = (x_i)_{i ∈ I} is a net in X then it is not in general true that Net_Tails(x•) is equal to x_• since, for instance, the domain of a net in X (i.e. the directed set I) may have any cardinality (so the class of nets in X isn't even a set) whereas the cardinality of the set of prefilters on X, which is a subset of ℘(℘(X)), is bounded above.

Proposition — Let ℬ be a prefilter on X and x ∈ X. Then

1. ℬ → x in (X, τ) if and only if Net_ℬ → x in (X, τ).
2. ℬ → x in (X, τ) if and only if Net_ℬ → x in (X, τ).

Proof

Recall that ℬ = Tails(Net_ℬ) and that if x_• is a net in X then x_• → x ⇔ Tails(x_•) → x, and that x is a cluster point of x_• ⇔ x is a cluster point of Tails(x_•). By using x_• := Net_ℬ and ℬ = Tails(Net_ℬ), it follows that ℬ → x ⇔ Tails(Net_ℬ) → x ⇔ Net_ℬ → x. It also follows that x is a cluster point of ℬ ⇔ x is a cluster point of Tails(Net_ℬ) ⇔ x is a cluster point of Net_ℬ

A prefilter ℬ on X is an ultra prefilter if and only if Tails(Net_ℬ) is an ultranet in X.

Non-equivalence of subnets and subfilters

Definition: Let N : (I, ≤) → X and S : (A, ≤) → X be nets in X where (I, ≤) and (A, ≤) are two directed sets (not necessarily related to each other).

We say that a map h : A → I is an order-preserving function if h(a) ≤ h(b) whenever a ≤ b for a, b ∈ A. We say that a subset R ⊆ I is cofinal in I if for every i ∈ I there exists some r ∈ R such that i ≤ r.

We say that the net S : (A, ≤) → X is a subnet of a net N : (I, ≤) → X if there exists an order-preserving function h : (A, ≤) → (I, ≤) such that h(A) is cofinal in I and S = N ∘ h.

One may show that if y_• = (y_a)_{a ∈ A} is a subnet of x_• = (x_i)_{i ∈ I} then Tails(x_•) ≤ Tails(y_•). However, in general there is no converse to. That is the following statement is in general false:

If ℬ and ℱ are prefilters such that ℬ ≤ ℱ then Net_ℱ is a subset of Net_ℬ.

It can be shown that there are prefilters ℬ and ℱ on the natural numbers X := ℕ such that ℬ ≤ ℱ but there is no order preserving map h : PointedSets( ℱ ) → PointedSets( ℬ ) such that the image of h is cofinal in its codomain and Net_ℱ = Net_ℬ ∘ h (in particular, let ℬ := { { 1 } ∪ ℕ_≥n : n ∈ ℕ } and let ℱ := { { 1 } } ∪ ℬ). This shows that nets and filters are not completely interchangeable and that there are relations that filters can express that nets can not. If, however, one was to use the following alternative definition of a subnet then this issue goes away while preserving all of the usual properties that the standard definition of "subnet" enjoys: say that y_• = (y_a)_{a ∈ A} is an alt-subnet of x_• = (x_i)_{i ∈ I} if Tails(x_•) ≤ Tails(y_•).

Characterizations in terms of filter bases

Throughout (X, τ) will be a topological space with X ≠ ∅.

Closure

If x ∈ X and S ⊆ X with S ≠ ∅ then the following are equivalent:

1. x ∈ cl S
2. x is a limit point of the prefilter { Y  } (i.e. { Y  } → x in (X, τ)).
3. There exists a prefilter ℱ ⊆ ℘(X) on X such that S ∈ ℱ and ℱ → x in (X, τ).
4. There exists a prefilter ℱ ⊆ ℘(S) on S such that ℱ → x in (X, τ).
5. x is a cluster point of the prefilter { Y  }.
6. The prefilter { Y  } meshes with the neighborhood filter 𝒩(x).
7. The prefilter { Y  } meshes with some (or equivalently, with every) filter base of 𝒩(x).

Closed

If S ⊆ X with S ≠ ∅ then the following are equivalent:

1. S is closed in (X, τ).
2. If x ∈ X and ℱ ⊆ ℘(S) is a prefilter on S such that ℱ → x in (X, τ), then x ∈ S.
3. If x ∈ X is such that the neighborhood filter 𝒩(x) meshes with { S  } then x ∈ S.
   • The proof of this characterization depends the ultrafilter lemma, which depends on the axiom of choice.

Hausdorff

The following are equivalent:

1. (X, τ) is Hausdorff.
2. Every prefilter on X converges to at most one point in X.
3. The above statement but with the word "prefilter" replaced by any one of the following: filter, ultra prefilter, ultrafilter.

Compact

The following are equivalent:

1. (X, τ) is a compact space.
2. Every prefilter on X has at least one cluster point in X.
3. For every filter ℱ on X there exists a filter ℛ on X such that ℱ ≤ ℛ and ℛ converges to some point of X.
4. For every prefilter ℱ on X there exists a prefilter ℛ on X such that ℱ ≤ ℛ and ℛ converges to some point of X.
5. Every maximal prefilter on X converges to at least one point in X.
6. The above statement but with the words "maximal prefilter" replaced by any one of the following: prefilter, filter, ultra prefilter, ultrafilter.

If (X, τ) is topological space and ℱ is the set of all complements of compact subsets of (X, τ), then ℱ is a filter on X if and only if (X, τ) is not compact.

Continuity

If f : X → Y is a map between topological spaces (X, τ_X) and (Y, τ_Y) then following are equivalent:

1. f : X → Y is continuous.
2. Whenever x ∈ X and ℱ is a prefilter on X such that ℱ → x in (X, τ_X) then f ( ℱ  ) → f (x) in (Y, τ_Y).
3. Whenever x ∈ X is a limit point of a prefilter ℱ on X then f (x) is a limit point of f ( ℱ  ) in (Y, τ_Y).
4. Any one of the above two statements but with the word "prefilter" replaced by any one of the following: filter.

Products

Suppose X_• = (X_i)_{i ∈ I} is a non-empty collection of non-empty topological spaces and that ℬ_• = (ℬ_i)_{i ∈ I} is a collection of prefilters where each ℬ_i is a prefilter on X_i. Then the product ℬ_• of these prefilters (defined above) is a prefilter on the product space ∏ X_•, which we will assume is endowed with the product topology. If x_• = (x_i)_{i ∈ I} ∈ ∏ X_•, then ℬ_• → x_• in ∏ X_• if and only if ℬ_i → x_i in X_i for every i ∈ I.

Completeness of topological vector spaces

Canonical uniformity

Every topological vector spaces (TVS) is a commutative topological group with identity under addition and the canonical uniformity of a TVS is defined entirely in terms of subtraction (and thus addition); scalar multiplication is not involved and no additional structure is needed. For this reason, we give definitions for an arbitrary additive commutative topological group (X, +) with identity 0.

Definition: The canonical uniformity on a topological vector space (X, τ) is defined to be the canonical uniformity of the commutative additive topological group (X, +, τ) (which we now define below).

Observe that a TVS does not need to be Hausdorff or satisfy any other conditions at all in order for the canonical uniformity to be defined.

Canonical uniformity on a topological group

We will henceforth assume that any topological group that we consider is an additive commutative topological group with identity element 0.

Notation: For any set X, let Δ := Δ_X := { (x, x) : x ∈ X  } denote the diagonal in X × X.

Notation: If (X, +) is a commutative group and S is a subset of X, then let          Δ(S) := Δ_X(S) := { (x, y) ∈ X × X : x - y ∈ S }.

Note that if 0 ∈ S then Δ_X(S) will contain the diagonal Δ_X.

Definition: If (X, +) is a commutative topological group and 𝒩 is a neighborhood basis of the identity element 0, then the canonical uniformity on X is the uniform structure on X that results from taking the upward closure of the following prefilter on X × X (which is called a base of vicinities of the diagonal Δ in X × X):

         { Δ(N) : N ∈ 𝒩 }

where Δ(N) := { (x, y) ∈ X × X : x - y ∈ N }.

This canonical uniformity is independent of the particular neighborhood basis of 0 that is chosen.

Having define a uniform structure on commutative topological groups, the notions of Cauchy nets, Cauchy filters, sequential completeness, and other notions are now defined via their usual definitions for uniform structures. However, for clarity, we review the relevant definitions again.

Cauchy filter bases and nets

Cauchy nets

Definition: A net x_• = (x_i)_{i ∈ I} in X is called a Cauchy net if for every neighborhood N of 0 in X, there exists some i₀ ∈ I such that x_i - x_j ∈ N for all i, j ≥ i₀ where i, j ∈ I. A Cauchy sequence is a Cauchy net that is a sequence.

Cauchy filter bases

Definition: If S is a subset of an additive group G and N is a set containing 0, then we say that S is N-small or small of order N if S - S ⊆ N.

Definition: A filter base ℬ on an additive topological group X called a Cauchy filter base if for every neighborhood N of 0 in X, there exists some B ∈ ℬ such that B - B ⊆ N.

The canonical uniformity is independent of the neighborhood basis 𝒩 that is chosen.

Complete topological group

Definition: A subset S of a topological group X is called complete if it satisfies any of the following equivalent conditions:

1. Every Cauchy prefilter 𝒞 ⊆ ℘(S) on S converges to at least one point of S.
   • If X is Hausdorff then every prefilter on S will converge to at most one point of X. But if X is not Hausdorff then a prefilter may converge to multiple points in X.
   • If X is not Hausdorff and if every Cauchy prefilter on S converges to some point of S, then S will be complete even if some or all Cauchy prefilters on S also converge to points(s) in X ∖ S. In short, there is no requirement that these Cauchy prefilters on S converge only to points in S. The same can be said of the convergence of Cauchy nets in S.
2. Every Cauchy net in S converges to at least one point of S;
3. Every Cauchy filter 𝒞 ⊆ ℘(S) on S converges to at least one point of S.
4. S is a complete subset (under the point-set topology definition of "complete") under the uniformity induced on S by the canonical uniformity;

Definition: A topological group X is called complete if X is complete as a subset of itself.

Examples and sufficient conditions

• Every Fréchet space, Banach space, and Hilbert space is a complete TVS.

Topologizing the set of filter bases and Top(X)

Starting with nothing more than a set X, one may topologize the set

ℙ := Prefilters(X)

of all filter bases on X with the Stone topology. We first define and describe the basic properties of this topology and then show how one may use it to easily topologize the set of all topologies on X; something is not easily done with nets in X.

To reduce confusion we will adhere to the following notational conventions:

• Lower case letters for elements x ∈ X.
• Upper case letters for subsets S ⊆ X.
• Upper case calligraphy letters for subsets 𝒜 ⊆ ℘(X).
• Upper case double-struck letters for subsets ℙ ⊆ ℘(℘(X)).

Observe that if R ⊆ S ⊆ X then { 𝒜 ∈ ℘(℘(X)) : R ∈ 𝒜^↑X } ⊆ { 𝒜 ∈ ℘(℘(X)) : S ∈ 𝒜^↑X }.

Definition: For every S ⊆ X, let      𝕆(S) := { 𝒜 ∈ ℙ : S ∈ 𝒜^↑X }

where note that 𝕆(X) = ℙ and 𝕆(∅) = ∅. One may show that for all R, S ⊆ X the following holds:

𝕆(R ∩ S) = 𝕆(R) ∩ 𝕆(S) ⊆ 𝕆(R) ∪ 𝕆(S) ⊆ 𝕆(R ∪ S)

where in particular, the equality 𝕆(R ∩ S) = 𝕆(R) ∩ 𝕆(S) shows that the collection of sets { 𝕆(S) : S ⊆ X } form a basis for a topology on ℙ, which we will henceforth assume ℙ carries. We will assume that any subset of ℙ carries the subspace topology.

Recall that every τ ∈ Top(X) induces a canonical map 𝒩_τ : X → Filter(X) defined by x ↦ 𝒩_τ(x). Clearly, 𝒩_τ : X → Filter(X) is injective if and only if τ is T₀ (i.e. a Kolmogorov space).

Notation: Let 𝒩_• : Top(X) → Func(X;  ℙ)      denote the map      τ ↦ 𝒩_τ.

Since 𝒩_• : Top(X) → Func(X;  ℙ) is clearly injective, to define a topology on Top(X) it suffices to define a topology on the range Im 𝒩_• := { 𝒩_τ : τ ∈ Top(X) }. So endow Func(X;  ℙ) with the topology of pointwise convergence (no topology on X is needed to do this) and endow Im 𝒩_• with the subspace topology. We've thus topologized Top(X).

We now describe some additional properties of the Stone topology. For any 𝕊 ⊆ ℙ and 𝒜 ∈ ℙ,

• 𝒜 belongs to the closure of 𝕊 in ℙ if and only if 𝒜 ⊆ ∪𝒮 ∈ 𝕊 𝒮^↑X.
• 𝕊 is a neighborhood of 𝒜 in ℙ if and only if there exists some A ∈ 𝒜 such that 𝕆(A) = { 𝒫 ∈ ℙ : A ∈ 𝒫^↑X } ⊆ 𝕊 (i.e. for all 𝒫 ∈ ℙ, if A ∈ 𝒫^↑X then 𝒫 ∈ 𝕊).

The set of ultrafilters on X (with the subspace topology) is a Stone space, meaning that it is compact, Hausdorff, and totally disconnected. The map β : X → UltraFilters(X) is a topological embedding whose image is a dense subset of UltraFilters(X).

For every τ ∈ Top(X), the map 𝒩_τ : (X, τ) → Im 𝒩_τ is continuous, closed, and open (where Im 𝒩_τ has the subspace topology inherited from ℙ). In addition, if 𝔉 : X → Filter(X) is a map such that x ∈ ker 𝔉(x) = ∩F ∈ 𝔉(x) F for every x ∈ X, then for every x ∈ X and every F ∈ 𝔉(x), 𝔉(F) is a neighborhood of 𝔉(x) in Im 𝔉 (where Im 𝔉 has the subspace topology inherited from ℙ).

See also

• Complete metric space – Metric geometry
• Filter (mathematics) – In mathematics, a special subset of a partially ordered set
• Filters in topology – Use of filters to describe and characterize all basic topological notions and results.
• The fundamental theorem of ultraproducts – Mathematical constructionPages displaying short descriptions of redirect targets
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Metric space – Mathematical space with a notion of distance
• Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
• Net (mathematics) – A generalization of a sequence of points
• Pseudometric space – Generalization of metric spaces in mathematics
• Stone–Čech compactification#Construction using ultrafilters – a universal map from a topological space X to a compact Hausdorff space βX, such that any map from X to a compact Hausdorff space factors through βX uniquely; if X is Tychonoff, then X is a dense subspace of βXPages displaying wikidata descriptions as a fallback
• Topological vector space – Vector space with a notion of nearness
• Uniform space – Topological space with a notion of uniform properties

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Category:Topological vector spaces