Free Filter

In mathematics, a filter on a set $X$ is a family $\mathcal$ of subsets such that:
# $X \in \mathcal$ and $\emptyset \notin \mathcal$
# if $A\in \mathcal$ and $B \in \mathcal$, then $A\cap B\in \mathcal$
# If $A,B\subset X,A\in \mathcal$, and $A\subset B$, then $B\in \mathcal$

A filter on a set may be thought of as representing a "collection of large subsets". Filters appear in order, model theory, set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal.

Filters were introduced by Henri Cartan in 1937 and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book ''Topologie Générale'' as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of filters from sets to arbitrary partially ordered sets. Specifically, a filter on a set is just a proper order filter in the special case where the partially ordered set consists of the power set ordered by set inclusion.

Preliminaries, notation, and basic notions

In this article, upper case Roman letters like $S \text X$ denote sets (but not families unless indicated otherwise) and $\wp(X)$ will denote the power set of $X.$ A subset of a power set is called (or simply, ) where it is if it is a subset of $\wp(X).$ Families of sets will be denoted by upper case calligraphy letters such as $\mathcal, \mathcal, \text \mathcal.$
Whenever these assumptions are needed, then it should be assumed that $X$ is non–empty and that $\mathcal, \mathcal,$ etc. are families of sets over $X.$

The terms "prefilter" and "filter base" are synonyms and will be used interchangeably.

Warning about competing definitions and notation

There are unfortunately several terms in the theory of filters that are defined differently by different authors.
These include some of the most important terms such as "filter."
While different definitions of the same term usually have significant overlap, due to the very technical nature of filters (and point–set topology), these differences in definitions nevertheless often have important consequences.
When reading mathematical literature, it is recommended that readers check how the terminology related to filters is defined by the author.
For this reason, this article will clearly state all definitions as they are used.
Unfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (for example, the notation for the set of all prefilters on a set) so in such cases this article uses whatever notation is most self describing or easily remembered.

The theory of filters and prefilters is well developed and has a plethora of definitions and notations, many of which are now unceremoniously listed to prevent this article from becoming prolix and to allow for the easy look up of notation and definitions.
Their important properties are described later.

Sets operations

The or in $X$ of a family of sets $\mathcal \subseteq \wp(X)$ is

and similarly the of $\mathcal$ is $\mathcal^ := \ = \bigcup_ \wp(B).$

Throughout, $f$ is a map and $S$ is a set.

Nets and their tails

A is a set $I$ together with a preorder, which will be denoted by $\,\leq\,$ (unless explicitly indicated otherwise), that makes $(I, \leq)$ into an () ; this means that for all $i, j \in I,$ there exists some $k \in I$ such that $i \leq k \text j \leq k.$ For any indices $i \text j,$ the notation $j \geq i$ is defined to mean $i \leq j$ while $i < j$ is defined to mean that $i \leq j$ holds but it is true that $j \leq i$ (if $\,\leq\,$ is antisymmetric then this is equivalent to $i \leq j \text i \neq j$).

A is a map from a non–empty directed set into $X.$
The notation $x_ = \left(x_i\right)_$ will be used to denote a net with domain $I.$

Warning about using strict comparison

If $x_ = \left(x_i\right)_$ is a net and $i \in I$ then it is possible for the set $x_ = \left\,$ which is called , to be empty (for example, this happens if $i$ is an upper bound of the directed set $I$).
In this case, the family $\left\$ would contain the empty set, which would prevent it from being a prefilter (defined later).
This is the (important) reason for defining $\operatorname\left(x_\right)$ as $\left\$ rather than $\left\$ or even $\left\\cup \left\$ and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality $\,<\,$ may not be used interchangeably with the inequality $\,\leq.$

Filters and prefilters

The following is a list of properties that a family $\mathcal$ of sets may possess and they form the defining properties of filters, prefilters, and filter subbases. Whenever it is necessary, it should be assumed that $\mathcal \subseteq \wp(X).$

Many of the properties of $\mathcal$ defined above and below, such as "proper" and "directed downward," do not depend on $X,$ so mentioning the set $X$ is optional when using such terms. Definitions involving being "upward closed in $X,$" such as that of "filter on $X,$" do depend on $X$ so the set $X$ should be mentioned if it is not clear from context.

$\textrm(X) \quad=\quad \textrm(X) \,\setminus\, \ \quad\subseteq\quad \textrm(X) \quad\subseteq\quad \textrm(X).$

\text X containing $\mathcal$ called the , and $\mathcal$ is said to this filter. This filter is equal to the intersection of all filters on $X$ that are supersets of $\mathcal.$ The –system generated by $\mathcal,$ denoted by $\pi(\mathcal),$ will be a prefilter and a subset of $\mathcal_.$ Moreover, the filter generated by $\mathcal$ is equal to the upward closure of $\pi(\mathcal),$ meaning $\pi(\mathcal)^ = \mathcal_.$ However, $\mathcal^ = \mathcal_$ if $\mathcal$ is a prefilter (although $\mathcal^$ is always an upward closed filter base for $\mathcal_$).
* A $\subseteq$–smallest (meaning smallest relative to $\subseteq$) filter containing a filter subbase $\mathcal$ will exist only under certain circumstances. It exists, for example, if the filter subbase $\mathcal$ happens to also be a prefilter. It also exists if the filter (or equivalently, the –system) generated by $\mathcal$ is principal, in which case $\mathcal \cup \$ is the unique smallest prefilter containing $\mathcal.$ Otherwise, in general, a $\subseteq$–smallest filter containing $\mathcal$ might not exist. For this reason, some authors may refer to the –system generated by $\mathcal$ as However, if a $\subseteq$–smallest prefilter does exist (say it is denoted by $\operatorname \mathcal$) then contrary to usual expectations, it is necessarily equal to " the prefilter generated by $\mathcal$" (that is, $\operatorname \mathcal \neq \pi(\mathcal)$ is possible). And if the filter subbase $\mathcal$ happens to also be a prefilter but not a -system then unfortunately, " the prefilter generated by this prefilter" (meaning $\pi(\mathcal)$) will not be $\mathcal = \operatorname \mathcal$ (that is, $\pi(\mathcal) \neq \mathcal$ is possible even when $\mathcal$ is a prefilter), which is why this article will prefer the accurate and unambiguous terminology of "the –system generated by $\mathcal$".

of a filter $\mathcal$ and that $\mathcal$ is a of $\mathcal$ if $\mathcal$ is a filter and $\mathcal \subseteq \mathcal$ where for filters, $\mathcal \subseteq \mathcal \text \mathcal \leq \mathcal.$
* Importantly, the expression "is a filter of" is for filters the analog of "is a sequence of". So despite having the prefix "sub" in common, "is a filter of" is actually the of "is a sequence of." However, $\mathcal \leq \mathcal$ can also be written $\mathcal \vdash \mathcal$ which is described by saying "$\mathcal$ is subordinate to $\mathcal.$" With this terminology, "is ordinate to" becomes for filters (and also for prefilters) the analog of "is a sequence of," which makes this one situation where using the term "subordinate" and symbol $\,\vdash\,$ may be helpful.

There are no prefilters on $X = \varnothing$ (nor are there any nets valued in $\varnothing$), which is why this article, like most authors, will automatically assume without comment that $X \neq \varnothing$ whenever this assumption is needed.

Basic examples

Named examples

• The singleton set $\mathcal = \$ is called the or It is the unique filter on $X$ because it is a subset of every filter on $X$; however, it need not be a subset of every prefilter on $X.$
• The dual ideal $$