Difference between revisions 19719 and 19729 on enwiki

In [[mathematics]], a '''filter''' is a special subset of a [[partially ordered set]]. A frequently used special case is the situation that the ordered set under consideration is just the [[power set]] of some set, ordered by set inclusion. Filters appear in [[order theory|order]] and [[lattice theory]], but can also be found in [[topology]]. The [[duality (order theory)|dual]] notion of a filter is an [[ideal (order theory)|ideal]].

Filters were introduced by [[Henri Cartan]] in [[1937]] and subsequently used by [[Bourbaki]] in their book ''[[Topologie Gènèrale]]''. An equivalent notion called [[net (topology)|net]] was developed in [[1922]] by [[Eliakim Hastings Moore|E. H. Moore]] and [[H. L. Smith]]. 

== General definition ==

A [[non-empty]] subset ''F'' of a partially ordered set (''P'',≤) is a '''filter''', if the following conditions hold:

# For every ''x'', ''y'' in ''F'', there is some element ''z'' in ''F'', such that ''z'' ≤ ''x'' and ''z'' ≤ ''y''. (''F'' is a '''filter base''')
# For every ''x'' in ''F'' and ''y'' in ''P'', ''x'' ≤ ''y'' implies that ''y'' is in ''F''. (''F is an upper set'')

A filter is '''proper''' if it is not equal to the whole set ''P''.

While the above definition is the most general way to define a filter for arbitrary posets, it was originally defined for [[lattice (order)|lattice]]s only. In this case, the above definition can be characterized by the following equivalent statement:
A non-empty subset ''F'' of a lattice (''P'',≤) is a filter, [[iff]] it is an upper set that is closed under finite meets ([[infimum|infima]]), i.e., for all ''x'', ''y'' in ''F'', we find that ''x'' ^ ''y'' is also in ''F''.

The smallest filter that contains a given element ''p'' is a '''principal filter''' and ''p'' is a '''principal element''' in this situation. The principal filter for ''p'' is just given by the set {''x'' in ''P'' | ''p'' ≤ ''x''} and is denoted by prefixing ''p'' with an upward arrow.

The dual notion of a filter, i.e. the concept obtained by reversing all ≤ and exchanging ^ with v, is '''ideal'''. Because of this duality, the discussion of filters usually boils down to the discussion of ideals. Hence, most additional information on this topic (including the definition of '''maximal filters''' and '''prime filters''') is to be found in the article on [[ideal (order theory)|ideals]]. There is a separate article on [[ultrafilter]]s.

== Filters of sets ==

An important special case of order filters are filters of sets, which are obtained by taking the powerset of a set ''S'' as a partial order, ordered by subset inclusion. Thus, a filter ''F'' on a [[set]] ''S'' is a set of [[Subset|subsets]] of ''S'' with the following properties:

# ''S'' is in ''F''. (''F is non-empty'')
# The empty set is not in ''F''. (''F is proper'')
# If ''A'' and ''B'' are in ''F'', then so is their intersection. (''F is closed under finite joins'')
# If ''A'' is in ''F'' and ''A'' is a subset of ''B'', then ''B'' is in ''F'', for all subsets ''B'' of ''S''. (''F is an upper set'')

Note that this definition is in absolute correspondence with the general notion introduced above, since the powerset clearly forms a lattice.

An important notion for filters of sets is that of a filter base. Given any subset ''T'' of ''P(S)'' such that the intersection of any finite subset of ''T'' is non-empty, there is a unique smallest filter ''F'' containing ''T'', called the filter generated by ''T''. If ''T'' is closed under finite intersections then ''F'' takes the simple form <math>F = \{ x : \exists y \in T \ y \subseteq x \}</math> and ''T'' is called a '''filter base''' for ''F''

A simple example of a filter is the set of all subsets of ''S'' that include a particular subset ''C'' of ''S''.  Such a filter is called the "principal filter" generated by ''C''. The [[Frechet filter|Fréchet filter]] on an infinite set ''S'' is the set of all subsets of ''S'' that have finite complement.

For any filter ''F'' on a set ''S'', the set function defined by
:<math>
m(A)=\left\{
\begin{matrix}
\,1 & \mbox{if }A\in F \\
\,0 & \mbox{if }S\setminus A\in F \\
\,\mbox{undefined} & \mbox{otherwise}
\end{matrix}
\right.
</math>
is finitely additive -- a "[[measure (mathematics)|measure]]" if that term is construed rather loosely.  Therefore the statement
:<math>\left\{\,x\in S: \varphi(x)\,\right\}\in F</math>
can be considered somewhat analogous to the statement that &phi; holds "almost everywhere".  That interpretation of membership in a filter is used (for motivation, although it is not needed for actual ''proofs'') in the theory of [[ultraproduct]]s in [[model theory]], a branch of [[mathematical logic]].

==Filters in topology==

Filters are used in [[topology]] and analysis. They are a good way of talking about convergence, in a manner similar to the role of [[sequence|sequences]] in a [[metric space]].

Given a point ''x'' the set of all neighbourhoods of ''x'' is a filter, <math>N_x</math>. A (proper) filter which is a superset of <math>N_x</math> is said to converge to ''x'', written <math>F \to x</math>. Note that if <math>F \to x</math> and <math>F \subseteq G</math> then <math>G \to x</math>. 

Given a filter ''F'' on a set ''X'' and a function <math>f : X \to Y</math>, the set <math> \{ f(A) : A \in F \} </math> forms a filter base for a filter which, in a slight abuse of notation, we denote by <math>f(F)</math>.

The following useful results hold:

# ''X'' is [[Hausdorff]] iff every filter on ''X'' has at most one limit.
# ''f'' is continuous at ''x'' iff <math> F \to x </math> implies <math>f(F) \to f(x)</math>
# ''X'' is compact iff every filter on ''X'' is a subset of a convergent filter.
# ''X'' is compact iff every ultrafilter on ''X'' converges.

==Filters in uniform spaces==

Given a [[uniform space]] X, a filter ''F'' on ''X'' is said to be Cauchy if for every ''U'' in the entourage, there is an <math>A \in F</math> with for every <math>x, y \in A \ \ (x, y) \in U</math>. In a metric space this takes the form  F is Cauchy if for every <math> \epsilon > 0 \ \ \exists A \in F \ \ \mathrm{diam}(A) < \epsilon </math>. ''X'' is said to be complete if every Cauchy filter converges.

Let <math>F \subseteq G , \ \ G \to x, \ F</math> Cauchy. Then <math>F \to x</math>. Thus every compact uniformity is complete. Further, a uniformity is compact iff it is complete and [[totally bounded]].

== See also ==

* [[Filtration (abstract algebra)]]

== References ==

*Cartan, H. (1937) "Thèorie des filtres". ''CR Acad. Paris'', '''205''', 595–598. 
*Cartan, H. (1937) "Filtres et ultrafiltres" ''CR Acad. Paris'', '''205''', 777–779

==External links==

[http://www.srcf.ucam.org/~drm39/filters.pdf An introductory account of the theory of filters in metric and topological spaces]

[[Category:Order theory]]

[[fr:Filtre (mathématiques)]]
[[de:Filter (Mathematik)]]==Dark ages==

When were the dark ages?
-----
The term 'dark ages' is not used by professional historians any more.

:That's largely missing the point, which is to communicate to users of wikipedia in the ways appropriate to how they are using it. Consider this excerpt: "Traditionally, the Middle Ages is said to begin when the West Roman empire formally ceased to exist..." Well, it's wrong. <i>Traditionally</i>, i.e. the way it used to be called, that was actually when the Dark Ages began. Traditionally, the Middle ages were considered to begin as early as Charlemagne or as late as Manzikert, depending on who was drawing the line and why (which varied with the geographical area in question). So, if the article is instructing people on modern usage, well and good (but the word "traditionally" should be changed to, say, "currently"). Conversely, the term "Dark Ages" should be handled properly for people like me, who have been using it as a way of explaining other things <i>in terms of people's ordinary understandings</i> - and for that, you have to avoid terms of art. PML.

: The more we learn about the Late Antiquity and the Early Middle Ages (note the preferred [[:Periodization|Periodization]]) the more clear it is that the 'Dark Ages' was a misnomer.  There was never a time in Europe when people didn't read and write, buy and sell, etc., etc.  There was a time when there was no particularly efficient central government in most parts of Europe, but if that's your definition of 'darkness,' then I can't help you.  The term 'dark' was originally used by Italians in the 15th century to refer to the time between themselves (obviously enlightened, a time of Rebirth [Rinascimento = Renaissance]) and the Greco-Romans.  Very self-satisfied of them to think of themselves as on one of two well-lit peaks with a dark valley between them, no? --MichaelTinkler

: Michael wrote: ''There was a time when there was no particularly efficient central government in most parts of Europe, but if that's your definition of 'darkness,' then I can't help you. '' He might have added: ''"and neither can the EEC in Brussels!"'' (Sorry - couldn't resist the easy shot - [[User:Tannin|Tannin]])

re: Perioditization and other terminology
--------

Michal is right about the usage of the term 'dark ages' -- a term I think was first used by Petrarch?  Another term that I have seen used (or rather, misused) in several places is 'feudalism.'  May I suggest that anyone using this term make sure they have first read the relevant articles by Peggy Brown and Susan Reynolds, and then use the term in a VERY qualified way?

[[User:JHK|JHK]]

==Feudalism==

I don't want to change this without some discussion, since I know there's a lot of academic debate on the subject, but there's something on the page that seems to suggest that the entirety of Europe was "feudal" for the entire Middle Ages.  Do we have room for (and would anyone mind) a more thourough discussion of "feudalism" and its permutations and variations, or should I go ahead and create a new page for it? -- Kate Secor
---------
go right ahead [[User:Chris mahan|Christopher Mahan]]
---------
Note that there is already a page about [[feudalism]]. --[[User:Eloquence|Eloquence]] 21:20 Nov 14, 2002 (UTC)
---------
Thanks, [[User:Eloquence|Eloquence]].  I put in a new link and am trying to figure out which of several new pages needs to get written first.  ;P -- [[User:Aiglet|Kate Secor]]

==Spelling of the adjective==

[[User:Tedius Zanarukando| Tedius Zanarukando]]  has edited the page so that it now reads:

(The corresponding adjective is spelt '''''medieval''''' in [[American English]] [influenced by French '''''mediéval'''''] 

Is there any evidence that the American spelling is influenced by French?  I would have thought that it was part of the general American simplification of spelling.  I can think of several instances of a British "ae" being shortened to an American "e". [[User:David Stapleton|David Stapleton]] 22:57, Nov 21, 2003 (UTC)

==Early medieval redirected==

why is "early medieval" redirected to medieval? I thought to do something about archaeological chronology, but this doesn't fit on the medieval page?

--[[User:Yak|Yak]] 11:55, Feb 26, 2004 (UTC)

==Message to all people editing history articles==

There seems to be a tendency for some editors to include the period between the Renaissance and the American Revolution, or at least the 18th century, into the "Middle Ages". I've seen articles discussing 17th or 18th century events after introducing them as "medieval". This is wrong with respect to the conventional historical periods. The period between the early 16th century and the late 18th century is generally referred to as "early modern", and its end as the "ancien régime" in some European contexts. [[User:David.Monniaux|David.Monniaux]] 10:29, 21 Aug 2004 (UTC)

: And, just for completeness, the article for this is at [[Early Modern period]]. --[[User:Joy|Joy <small><small>&#91;shallot&#93;</small></small>]]

== Middle Ages WikiProject ==

[[User:Stbalbach]] has set up a [[Wikipedia:WikiProject Middle Ages|Middle Ages WikiProject]], if anyone is interested. [[User:Adam Bishop|Adam Bishop]] 07:16, 17 Dec 2004 (UTC)

==China edit==

From China's Middle Ages period, I removed the following text: ' Needham points out that the [[lever]], a [[simple machine]], was not implemented with straight rods in China, but rather had corners -- a clear misunderstanding of the principle. Perhaps the problem was compounded by the sense that China had the greatest civilization on earth, at the time.' -- I think it's too much point of view and too detailed for this article. -- [[User:Cugel|Cugel]] 08:57, Mar 4, 2005 (UTC)