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
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.


 * Sets operations


 * Definition: The upward closure or isotonization of a collection $℘(X) &thinsp;:=&thinsp; { S : S ⊆ X&thinsp;}$ of subsets of $X$ is



Nets and topologies

 * Directed sets and nets notation.


 * Definition: A directed set is a set $X$ together with a preorder, which we will assume is denoted by $℘^{2}(X) &thinsp;:=&thinsp; ℘(℘(X))$ (unless otherwise specified), that makes $℘(X)$ into an upward directed set, which means that for every $Top(X)$, there exists some $Filters(X)$ such that $PreFilters(X) = FilterBases(X)$ and $UltraFilters(X)$. We define $Func(X; Y)$ to mean $ℬ$. A net in $X$ is a map from a directed set into $X$.


 * Topology notation

If $ℬ^{↑} &thinsp;:=&thinsp; ℬ^{↑X} &thinsp;:=&thinsp; { S ⊆ X : B ⊆ S for some B ∈ ℬ&thinsp;} &thinsp;=&thinsp; ∪ B ∈ ℬ { S : B ⊆ S ⊆ X&thinsp;}$ is a topology on $X$ then we may use the following notation.

Finer, coarser, subordinate, product
The following definition of $ker ℬ &thinsp;:=&thinsp; ∩ B ∈ ℬ B$ allows for the filter equivalent of "subsequence." It will also be used to define convergence. The definition of $ℬ ∩ { S } &thinsp;:=&thinsp; { B ∩ S : B ∈ ℬ&thinsp;}$ meshes with $S ⊆ X$ is used in Topology to define cluster points.



 
 * Note that if $ℬ$ then $S ∖ ℬ &thinsp;:=&thinsp; { S ∖ B : B ∈ ℬ&thinsp;}$. Thus $S ⊆ X$ is always true, for any $ℬ^{↓} &thinsp;:=&thinsp; ℬ^{↓X} &thinsp;=&thinsp; ∪ B ∈ ℬ { S : S ⊆ B&thinsp;}$.
 * $S ⊆ X$ is transitive: $𝒜 + ℬ &thinsp;:=&thinsp; { A + B : A ∈ 𝒜, B ∈ ℬ&thinsp;}$ and $s ℬ &thinsp;:=&thinsp; { s B : B ∈ ℬ&thinsp;}$ implies $f (ℬ) &thinsp;:=&thinsp; { f (B) : B ∈ ℬ&thinsp;}$.
 * $f : X → Y$ is reflexive: $ℬ$ is always true.






 * The relation $f&thinsp;^{-1} (𝒞) &thinsp;:=&thinsp; { f&thinsp;^{-1} (C) : C ∈ 𝒞&thinsp;}$ is not antisymmetric; that is, $f : X → Y$ and $𝒞 ⊆ ℘(Y)$ does not necessarily imply $ℬ$; not even if both $Im f &thinsp;:=&thinsp; f (X) &thinsp;=&thinsp; { f (x) : x ∈ X&thinsp;}$ and $f : X → Y$ are filter bases. However, $f$ is transitive and reflexive so this definition does indeed define an equivalence relation.







 
 * If $S^{↑X} &thinsp;:=&thinsp; { S }^{↑X}$ then we write $S ⊆ X$ to mean $𝒜 ⊓ ℬ &thinsp;:=&thinsp; 𝒜 ∩ ℬ &thinsp;:=&thinsp; { A ∩ B : A ∈ 𝒜, B ∈ ℬ&thinsp;}$.
 * If $≤$ and $(I, ≤)$ are filter bases on $X$ then $i, j ∈ I$ and $k ∈ I$ mesh if and only if there exists a filter base $i ≤ k$ on $X$ such that $j ≤ k$ and $j ≥ i$, or equivalently, if and only if $i ≤ j$ is a filter base on $Y$.

 

 

Filters and filter bases
We now define properties that a collection $I_{≥ i} &thinsp;:=&thinsp; { j ∈ I : i ≤ j&thinsp;}$ may have.

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

We now list some more properties that $i ∈ I$ may have.

Examples and properties

 * Images and preimages of filter bases

 <li> If $(I, ≤)$ is a prefilter (resp. ultra prefilter) on $X$ then $I_{> i} &thinsp;:=&thinsp; { j ∈ I : i ≤ j, &thinsp;j ≠ i&thinsp;}$ is a prefilter (resp. ultra prefilter) on $S$. </li> <li> If $i ∈ I$ is a filter on $X$ then $(I, ≤)$, and $f (I_{≥ i}) &thinsp;=&thinsp; { f (j) : i ≤ j, &thinsp;j ∈ I&thinsp;}$ is a filter on $X$ if and only if $s$ is surjective. </li> <li> If $i ∈ I$ is a prefilter on $f$ then $f : (I, ≤) → X$ is a prefilter on $f$ and moreover, $f (I_{> i}) &thinsp;=&thinsp; { f (j) : i ≤ j, &thinsp;j ≠ i, &thinsp;j ∈ I&thinsp;}$. </li> <li> If $i ∈ I$ is an ultrafilter on $I$ and $X$ is surjective then $f : (I, ≤) → X$ is an ultrafilter on $X$. </li> <li> A map $(x_{•})_{i} &thinsp;:=&thinsp; x_{≥ i} &thinsp;:=&thinsp; { x_{j} : i ≤ j, &thinsp;j ∈ I&thinsp;}$ is injective if and only if whenever $i ∈ I$ is a filter on $I$ then $x_{•} = (x_{i})_{i ∈ I}$ is a filter on $i$. </li> <li> If $I$ is a bijection then $x_{•}$ is a prefilter (resp. filter, ultrafilter) on $i$ if and only if the same is true of $x_{> i} &thinsp;:=&thinsp; { x_{j} : i ≤ j, &thinsp;j ≠ i, &thinsp;j ∈ I&thinsp;}$ on $f$. </li> <li> If $i ∈ I$ is a filter base on $i$ then $x_{•} = (x_{i})_{i ∈ I}$ is a filter base on $f$ if and only if $x_{•}$, in which case $Tails(x_{•}) &thinsp;:=&thinsp; (x_{i}) i &thinsp;:=&thinsp; x_{≥ •} &thinsp;:=&thinsp; { x_{≥ i} : i ∈ I&thinsp;}$. </li> <li> If $x_{•} = (x_{i})_{i ∈ I}$ is a filter on $i$ then $x_{•}$ is a filter base on $i$ but it may fail to be a filter on $i$ even if $X$ is surjective. </li> </ul>


 * Properties of ultrafilters

<ul> <li>A free ultrafilter on $S$ exists if and only if $x$ is infinite. <li>An ultra prefilter generates an ultrafilter.</li> <li>Every filter is equal to the intersection of all ultrafilters containing it.</li> <li>If $x_{•}$ then $TailsFilter(x_{•}) &thinsp;:=&thinsp; Tails(x_{•})^{↑X}$ can be extended to a free ultrafilter if and only if the intersection of any finite collection of elements of $x_{•} = (x_{i})_{i ∈ I}$ is infinite.</li> <li>A principal filter $x_{•}$ on $S$ (i.e. a filter that satisfies $τ$) is an ultrafilter if and only if $τ (S) &thinsp;:=&thinsp; { O ∈ τ : S ⊆ O&thinsp;}$ is a singleton set. All other ultrafilters on $S ⊆ X$ are free (i.e. $(X, τ)$).</li> </ul>
 * A filter $τ (x) &thinsp;:=&thinsp; { O ∈ τ : x ∈ O&thinsp;}$ 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 $(X, τ)$ for some $𝒩_{τ}(S) &thinsp;:=&thinsp; 𝒩(S) &thinsp;:=&thinsp; τ (S)^{↑X}$).</li>

$X$

$X$


 * Examples of filter bases and filters

<ul> <li> 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. </li> <li> If $S ⊆ X$ is a filter base then the trace of $(X, τ)$ on $i$ is a filter base if $𝒩_{τ}(x) &thinsp;:=&thinsp;𝒩(x) &thinsp;:=&thinsp; τ (x)^{↑X}$ for all $x ∈ X$. </li> </ul>
 * However, if $A$ contains at least 2 distinct elements then there exist filters $(X, τ)$ and $𝒩_{τ} : X → Filter(X)$ on $B$ for which there does not exist a filter $x ↦ 𝒩_{τ}(x)$ on $X$ that contains both $τ$ and $X$.

Properties
<ul> <li> If $𝒜 ≤ ℬ$ and $𝒜$ is a filter base then $ℬ$ and $𝒜 ≤ ℬ$ mesh (see footnote for proof). </li> </ul>

Filter bases on topological spaces
Throughout, $ℬ ⊢ 𝒜$ is a topological space.


 * A note on intuition

Suppose that $ℬ$ is a non-principal filter on an infinite set $C$. $𝒜$ has one "upward" property (that of being closed upward) and one "downward" property (that of being directed downwards under set inclusion). Starting with any $𝒜$, there always exists some $ℬ$ that is a proper subset of $ℬ$; this may be continued ad infinitum to get $𝒜$ in $A ∈ 𝒜$ with each $B ∈ ℬ$ being a proper subset of $A ∈ 𝒜$. The same is not true going "upward", for if $B ∈ ℬ$ then there is no set in $B ⊆ A$ 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
Recall that the closure of a set $𝒜 ⊆ ℬ^{↑X}$ is equal to the union of $B$ together with the set of all limit points of $X$.


 * Limit and cluster points of filters

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

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

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

Every limit point of a filter base $ℬ$ is a cluster point of $≤$, since if $X$ is a limit point of a filter base $𝒜 ≤ ℬ$ then $ℬ ≤ 𝒜$ and $ℬ = 𝒜$ mesh and thus $X$ is a cluster point of $𝒜$. If $ℬ$ is an ultra prefilter on $X$ and $≤$, then $X$ is a cluster point of $𝒜 ◅ ℬ$ if and only if $ℬ ▻ 𝒜$ $𝒜$. The set of all cluster points of a filter base $ℬ$ in a topological space $x$ is a closed subset of $x$ and moreover, $𝒜$.

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

Filters and nets
The relation $A ⊆ B$ is of fundamental importance to applying filters to topology. We may use the $(X ∖ 𝒜) ≤ (X ∖ ℬ)$ 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.

<ul> <li> The notion of "$𝒜 # ℬ$ is subordinate to $𝒜$" (written $ℬ$) is for filters and filter bases what "$A ∩ B ≠ ∅$ is a subsequence of $A ∈ 𝒜$" is for sequences (and nets). </li> <li> If $B ∈ ℬ$ is a net in a topological space $X$, $S ⊆ X$ is the set of its tails, and $S # ℬ$ is the neighborhood filter at a point ${ S } # ℬ$, then $𝒜$ in $Y$ if and only if $ℬ$. </li> </ul>
 * Indeed, if we let $𝒜$ denotes the set of tails of $ℬ$ and $𝒞$ denotes the set of tails of the subsequence $𝒜 ≤ 𝒞$, then $ℬ ≤ 𝒞$ (i.e. $𝒜 ∩ ℬ$) is true but $𝒜$ is in general false.

A net $ℬ$ in $X$ is an ultranet if and only if $𝒜$ is an ultra prefilter.


 * Prefilters as nets

Observe that if $ℬ$ is a prefilter on $Y$ then $X_{•} = (X_{i})_{i ∈ I}$ is a directed set so if we desire a map of the form $ℬ_{•} = (ℬ_{i})_{i ∈ I}$ then the obvious choice is the assignment $ℬ_{i} ⊆ ℘(X_{i})$.

One may show that if $ℬ_{•}$ is a prefilter on $f$ then $ℬ_{i}$ is a net in $X$, and that the prefilter associated with $∏ ℬ_{•} := ∏ i ∈ I ℬ_{i}$ is $∏ i ∈ I S_{i}$. This would not necessarily be true had $∏ X_{•} := ∏ i ∈ I X_{i}$ been defined on a proper subset of $S_{i} = X_{i}$. For instance, if $X$ has at least two distinct elements, $i ∈ I$ is the indiscrete filter on $X$, $S_{i} ≠ X_{i}$ is arbitrary, and $S_{i} ∈ ℬ_{i}$ been defined on the singleton set $ℬ ⊆ ℘(X)$, then the prefilter associated with $ℬ$ would be the principal prefilter $∅ ∉ ℬ$ rather than $∅ ∈ ℬ$ (where note that $B, C ∈ ℬ$ is the unique minimal filter on $f$ whereas $B ∩ C ∈ ℬ$ generates a maximal/ultrafilter on $Y$).

However, if $B, C ∈ ℬ$ is a net in $Y$ then it is not in general true that $A ∈ ℬ$ is equal to $A ⊆ B ∩ C$ since, for instance, the domain of a net in $X$ (i.e. the directed set $f$) 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 $Y$, which is a subset of $ℬ$, is bounded above.

$Y$

$X$

A prefilter $B ≤ A$ on $Y$ is an ultra prefilter if and only if $A ⊆ B$ is an ultranet in $X$.


 * Non-equivalence of subnets and subfilters

One may show that if $ℬ$ is a subnet of $B_{1}, ..., B_{n} ∈ ℬ$ then $∅ ≠ B_{1} ∩ ⋅⋅⋅ ∩ B_{n}$. However, in general there is no converse to. That is the following statement is in general false:


 * If $ℬ = ℬ^{↑X}$ and $B ∈ ℬ$ are prefilters such that $B ⊆ C ⊆ X$ then $C ∈ ℬ$ is a subset of $ℬ^{↑X}$.

It can be shown that there are prefilters $ℬ^{↑X}$ and $ℬ$ on the natural numbers $S ⊆ X$ such that $B ∈ ℬ$ but there is no order preserving map $B ⊆ S$ such that the image of $X$ is cofinal in its codomain and $B ⊆ X ∖ S$ (in particular, let $B ∩ S$ and let $∅$). 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 $ℬ ⊆ ℘(X)$ is an alt-subnet of $ℬ$ if $ℬ ≠ ∅$.

Characterizations in terms of filter bases
Throughout $ℬ ≠ ∅$ will be a topological space with $𝔽$.


 * Closure

If $𝔽$ and $ℱ$ with $ℬ$ then the following are equivalent: <ol> <li>$ℬ ≤ ℱ$</li> <li>$f$ is a limit point of the prefilter $ℱ$ (i.e. $ℬ ≤ ℱ$ in $ℬ ⊆ ℱ$).</li> <li>There exists a prefilter $ℬ ≠ ∅$ on $X$ such that $℘(X)$ and $ℬ$ in $ℬ^{↑X}$.</li> <li>There exists a prefilter $ℬ$ on $X$ such that $ℬ$ in $ℱ$.</li> <li>$X$ is a cluster point of the prefilter $ℬ$.</li> <li>The prefilter $ℱ = ℬ^{↑X}$ meshes with the neighborhood filter $ℬ$.</li> <li>The prefilter $ℱ$ meshes with some (or equivalently, with every) filter base of $ℬ ≠ ∅$.</li> </ol>


 * Closed

If $ℬ$ with $ℬ$ then the following are equivalent: <ol> <li>$X$ is closed in $ℬ$.</li> <li>If $ℬ$ and $ℬ ≠ ∅$ is a prefilter on $$ such that $ℬ$ in $ℬ$, then $𝒟$.</li> <li>If $ℬ ≤ 𝒟$ is such that the neighborhood filter $𝒟 ≤ ℬ$ meshes with $ℬ$ then $ℬ$. </ol>
 * The proof of this characterization depends the ultrafilter lemma, which depends on the axiom of choice.</li>


 * Hausdorff

The following are equivalent: <ol> <li>$S ⊆ X$ is Hausdorff.</li> <li>Every prefilter on $$ converges to at most one point in $X$.</li> <li>The above statement but with the word "prefilter" replaced by any one of the following: filter, ultra prefilter, ultrafilter.</li> </ol>


 * Compact

The following are equivalent: <ol> <li>$B ∈ ℬ$ is a compact space.</li> <li>Every prefilter on $X$ has at least one cluster point in $X$.</li> <li>For every filter $B ⊆ S$ on $X$ there exists a filter $B ⊆ X ∖ S$ on $X$ such that $ℬ$ and $ℬ$ converges to some point of $S$.</li> <li>For every prefilter $S ⊆ X$ on $X$ there exists a prefilter $S ∈ ℬ$ on $X$ such that $X ∖ S ∈ ℬ$ and $ℬ$ converges to some point of $x$.</li> <li>Every maximal prefilter on $S$ converges to at least one point in $x$.</li> <li>The above statement but with the words "maximal prefilter" replaced by any one of the following: prefilter, filter, ultra prefilter, ultrafilter.</li> </ol>

If $S ⊆ X$ is topological space and $S$ is the set of all complements of compact subsets of $S ∈ ℬ$, then $x$ is a filter on $S$ if and only if $X ∖ S ∈ ℬ$ is not compact.


 * Continuity

If $ℬ ∪ (X ∖ ℬ) = ℘(X)$ is a map between topological spaces $ℬ ∩ (X ∖ ℬ) = ∅$ and $ℬ$ then following are equivalent: <ol> <li>$S_{1}, ..., S_{n}$ is continuous.</li> <li>Whenever $S_{1} ∪ ⋅⋅⋅ ∪ S_{n} ∈ ℬ$ and $S_{i} ∈ ℬ$ is a prefilter on $S$ such that $R, S ⊆ X$ in $R ∪ S ∈ ℬ$ then $R ∈ ℬ$ in $S ∈ ℬ$.</li> <li>Whenever $R, S ⊆ X$ is a limit point of a prefilter $R ∩ S = ∅$ on $S$ then $R ∪ S ∈ ℬ$ is a limit point of $R ∈ ℬ$ in $S ∈ ℬ$.</li> <li>Any one of the above two statements but with the word "prefilter" replaced by any one of the following: filter.</li> </ol>


 * Products

Suppose $ℬ$ is a non-empty collection of non-empty topological spaces and that $ℱ$ is a collection of prefilters where each $ℬ ⊆ ℱ$ is a prefilter on $ℬ = ℱ$. Then the product $ℬ$ of these prefilters (defined above) is a prefilter on the product space $ℱ$, which we will assume is endowed with the product topology. If $ℬ$, then $ℬ ≠ ∅$ in $ℬ$ if and only if $ℬ$ in $S ⊆ X$ for every $S ∈ ℬ$.

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 $X ∖ S ∈ ℬ$ with identity $ℬ$.

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.

Note that if $ℬ$ then $ℬ$ will contain the diagonal $ker ℬ = ∅$.

This canonical uniformity is independent of the particular neighborhood basis of $ℬ$ 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


 * Cauchy filter bases

The canonical uniformity is independent of the neighborhood basis $ker ℬ ≠ ∅$ that is chosen.

Examples and sufficient conditions
<ul> <li>Every Fréchet space, Banach space, and Hilbert space is a complete TVS.</li> </ul>

Topologizing the set of filter bases and Top(X)
Starting with nothing more than a set $S$, one may topologize the set



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 $N$.

To reduce confusion we will adhere to the following notational conventions: <ul> <li>Lower case letters for elements $ℬ$.</li> <li>Upper case letters for subsets $x ∈ ker ℬ$.</li> <li>Upper case calligraphy letters for subsets $ker ℬ ∈ ℬ$.</li> <li>Upper case double-struck letters for subsets $x ∈ X$.</li> </ul>

Observe that if ${ x } = ker ℬ ∈ ℬ$ then ${ x }^{↑X}$.

where note that $ℱ$ and $ℱ = { x }^{↑X}$. One may show that for all $ℬ = { X&thinsp;}$ the following holds:



where in particular, the equality $B ∈ ℬ$ shows that the collection of sets $U ∈ ℬ$ form a basis for a topology on $U + U ⊆ B$, which we will henceforth assume $ℬ$ carries. We will assume that any subset of $ℬ ⊆ ℬ + ℬ$ carries the subspace topology.

Recall that every $ℬ$ induces a canonical map $f (ℬ)$ defined by $ℬ$. Clearly, $f (ℬ)^{↑Y} := { S ⊆ Y : f&thinsp;^{-1} (S) ∈ ℬ&thinsp;}$ is injective if and only if $f (ℬ)$ is $ℬ$ (i.e. a Kolmogorov space).

Since $f&thinsp;^{-1} (f (ℬ))$ is clearly injective, to define a topology on $f&thinsp;^{-1} (f (ℬ)) ≤ ℬ$ it suffices to define a topology on the range $ℬ$. So endow $f (ℬ)$ with the topology of pointwise convergence (no topology on $x$ is needed to do this) and endow $f : X → Y$ with the subspace topology. We've thus topologized $ℱ$.

We now describe some additional properties of the Stone topology. For any $f&thinsp;^{-1} (ℱ&thinsp;)$ and $ℬ$, <ul> <li> $f (ℬ)$ belongs to the closure of $𝒞$ in $f&thinsp;^{-1} (𝒞)$ if and only if $∅ ∉ f&thinsp;^{-1} (𝒞)$. </li> <li> $𝒞 ≤ f (f&thinsp;^{-1} (𝒞))$ is a neighborhood of $𝒞$ in $f&thinsp;^{-1} (𝒞)$ if and only if there exists some $ℱ$ such that $x ∈ X$ (i.e. for all $ker ℱ = { x } ∈ ℱ$, if $x ∈ X$ then $∅≠ ℬ ⊆ ℘(X)$). </li> </ul>

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

For every $𝒫$, the map $ker 𝒫 ∈ 𝒫$ is continuous, closed, and open (where $ker 𝒫$ has the subspace topology inherited from $𝒰$). In addition, if $ker 𝒰 = ∅$ is a map such that $𝒰$ for every $𝒰$, then for every $𝒰$ and every $ker 𝒰 ≠ ∅$, $𝒰$ is a neighborhood of $𝒰$ in $x ∈ X$ (where $ker 𝒰 = { x } ∈ 𝒰$ has the subspace topology inherited from $x ∈ X$).