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Jan 23, 2003, 9:31:19 AM1/23/03

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I've recently been flicking through Bourbaki's 'General Topology', and

it made me wonder about Filters. I've only seen them used in two things

- both analysis; I first encountered them in the construction of the

hyperreals for non-standard analysis: One takes the set of all sequences

of real numbers, and defines an equivalence relation on them and an

order on the set of equivalence classes using a free ultrafilter on the

set of natural numbers. Secondly they are used as a type of generalised

sequence in topological spaces - hence the connection to Bourbaki. I'm

more used to nets, but I can see the advantages that filters have as a

way of looking at it.

it made me wonder about Filters. I've only seen them used in two things

- both analysis; I first encountered them in the construction of the

hyperreals for non-standard analysis: One takes the set of all sequences

of real numbers, and defines an equivalence relation on them and an

order on the set of equivalence classes using a free ultrafilter on the

set of natural numbers. Secondly they are used as a type of generalised

sequence in topological spaces - hence the connection to Bourbaki. I'm

more used to nets, but I can see the advantages that filters have as a

way of looking at it.

They don't however seem like an inherently analytical idea - I'd expect

them to have uses is other areas of maths. Are there any examples of such?

Also, historically who was the first to introduce the idea of a filter?

And why is it named a filter?

Thanks,

David

(E-mail adress altered to prevent spam - remove the obvious extra from

it to e-mail me.)

Jan 23, 2003, 11:18:20 AM1/23/03

to

In article <3E2FFCB7...@spoon.cam.ac.uk>,

David R MacIver <dr...@spoon.cam.ac.uk> wrote:

>I've recently been flicking through Bourbaki's 'General Topology', and

>it made me wonder about Filters. I've only seen them used in two things

>- both analysis; I first encountered them in the construction of the

>hyperreals for non-standard analysis: One takes the set of all sequences

>of real numbers, and defines an equivalence relation on them and an

>order on the set of equivalence classes using a free ultrafilter on the

>set of natural numbers. Secondly they are used as a type of generalised

>sequence in topological spaces - hence the connection to Bourbaki. I'm

>more used to nets, but I can see the advantages that filters have as a

>way of looking at it.

>

>They don't however seem like an inherently analytical idea - I'd expect

>them to have uses is other areas of maths. Are there any examples of such?

David R MacIver <dr...@spoon.cam.ac.uk> wrote:

>I've recently been flicking through Bourbaki's 'General Topology', and

>it made me wonder about Filters. I've only seen them used in two things

>- both analysis; I first encountered them in the construction of the

>hyperreals for non-standard analysis: One takes the set of all sequences

>of real numbers, and defines an equivalence relation on them and an

>order on the set of equivalence classes using a free ultrafilter on the

>set of natural numbers. Secondly they are used as a type of generalised

>sequence in topological spaces - hence the connection to Bourbaki. I'm

>more used to nets, but I can see the advantages that filters have as a

>way of looking at it.

>

>They don't however seem like an inherently analytical idea - I'd expect

>them to have uses is other areas of maths. Are there any examples of such?

Yes. Filters and ultrafilters are very common in algebra, and a

particular incarnation of filters form (together with "ideals") one of

the fundamental substructures of lattices. If L is a lattice, then a

filter of L is a nonempty subset which is upward closed and closed

under meets (so, if x is in L and x<=y, then y is in l; and if x,y are

in L, then x ^ y is in L; you can see the analogies with the filters

defined in Bourbaki).

Filters and ultrafilters are used in general algebra to define certain

kinds of quotients: suppose you are working in the category of all

Omega-algebras of some fixed type, and let {A_i}_{i in I}. We can form

the product P of all A_i. Now let F be any filter in the power set of

I. We define an equivalence relation on P by:

(a_i) ~ (b_i) if and only if

{i in I; a_i=b_i} is in F.

Then this equivalence relation is actually a congruence (a subalgebra

of PxP which is also an equivalence relation), so we can mod out by

the congruence to get a new structure P/~, sometimes called a

"filtered product" of the A_i.

When F is an ultrafilter, this leads to a very interesting structure

(assuming Zorn's Lemma, anyway, so you can have nontrivial

ultrafilters on an infinite set I). For example, suppose you have an

infinite family as above, and let F be a nontrivial ultrafilter; let

Q=P/~, as above. This is called an "ultraproduct;" in Gratzer's

_Universal Algebra_, this is defined dually by taking a prime ideal I

(the dual of an ultraproduct), and taking the relation to be

(a_i) ~ (b_i) if and only if

{i in I; a_i<>b_i} is in I

and then called a "prime product."

The interesting property of the ultraproduct \prod_{F}{A_i} is that a

formula is true in the ultraproduct if and only if the set of indices

such that the formula is true in A_i lies in F.

So, for example, if F is a nontrivial ultrafilter, then F contains the

complement of every finite subset of I. So if a formula is true in

almost all A_i, then it is true on \prof_{F}{A_i}. An example from my

Qual: let {A_i} be the family of all finite prime fields (i.e., Z/pZ

for each prime p), and let F be a nontrivial ultrafilter.

The ultraproduct is a field, because the formula

"for all x exists y such that x*y=1" holds in all A_i. And the product

is a field of characteristic zero: because for each prime p, we can

take the formula

"there exists x such that not(x+x+...+x =0)"

where we have added x to itself p times, and this is true in all but

one of the A_i.

======================================================================

"It's not denial. I'm just very selective about

what I accept as reality."

--- Calvin ("Calvin and Hobbes")

======================================================================

Arturo Magidin

mag...@math.berkeley.edu

Jan 23, 2003, 11:56:38 AM1/23/03

to

> Also, historically who was the first to introduce the idea of a filter?

> And why is it named a filter?

There is an abstract published by Garrett Birkhoff describing (what we

would call today) convergence of a filter base. Perhaps this

publication pre-dates any publication on this by the French school.

When young Garrett's father (G. D. Birkhoff) saw it, he remarked that

it looks sort of like something in papers of Moore and Smith. So

Garrett's subsequent work on convergence in general topology was done

with the Moore-Smith scheme. [I heard this chronology from Garrett

Birkhoff.]

Anyway, when looking for "first introduction", this Birkhoff abstract

is a candidate. As I recall, back in the 1930's, abstracts for papers

delivered at meetings were published in the Proceedings of the A.M.S.?

If so, that is where to look for this.

--

G. A. Edgar http://www.math.ohio-state.edu/~edgar/

Jan 24, 2003, 8:16:12 AM1/24/03

to

David R MacIver <dr...@spoon.cam.ac.uk> wrote:

>[...] > Also, historically who was the first to introduce the idea of a filter?

> And why is it named a filter?

> And why is it named a filter?

The most obvious seems to be Cartan (1938),

but also Vietoris (1922).

See

<http://informatik.uibk.ac.at/users/reitberg/Vietoris_Tietze.pdf>

for more.

Marc

Jan 24, 2003, 4:02:28 PM1/24/03

to

David R MacIver <dr...@spoon.cam.ac.uk>

[sci.math Jan 23 2003 10:30:33:000AM]

http://mathforum.org/discuss/sci.math/m/476206/476206

[sci.math Jan 23 2003 10:30:33:000AM]

http://mathforum.org/discuss/sci.math/m/476206/476206

wrote (in part):

> Also, historically who was the first to introduce the idea

> of a filter? And why is it named a filter?

Below are what some references I looked at have to say

about this.

--------------------------------------------------------------

1. Manya Raman, "Understanding Compactness: A Historical

Perspective", Masters Thesis (UC-Berkeley, 1997), 34 pages.

http://socrates.berkeley.edu/~manya/compact/

(From pages 22-23) Another generalization of sequence is

filter, a notion suggested by Cartan in 1937. There is a

lot of interesting history and folklore surrounding nets

and filters. It turns out that nets are more popular in

the U.S. and filters in Europe (Temple, p. 114). [fn-8]

[(Renfro) See item #4 below.] One might think that this

division comes from the fact that Moore and Smith were

American and Cartan was French (Kelly, p. 83). However,

it turns out that Smith actually independently discovered

filters as an attempt to explain what was lacking in the

theory of nets that he and Moore proposed. Moreover, the

idea behind filters was actually foreshadowed by Riesz in

1909 when he provided axioms for topology based on limit

points instead of metrics. Riesz defines a concept called an

"ideal" which is essentially the same as what we now call an

untrafilter (Temple, p. 111).

Footnotes on pp. 22-23 -->

(fn-7) Incidentally, Kelley, who first coined the term "net",

had considered using the term "way" so the analog of

a subsequence would be "subway." McShane also proposed

the term "stream" for net since he thought it was

intuitive to think of the relation of the directed set

as "being downstream from" (McShane, p. 282).

[(Renfro) See McShane's footnote in item #3 below.]

(fn-8) It may be surprising for some that there is any

difference between nets and filters. For all practical

purposes there isn't. In fact, an exercise in Kelly

shows there is a dictionary mapping between them

(i.e. given a net you can find a filter, and vice versa)

(Kelly, p. 83). But there is a subtle distinction for a

particular type of limit found in the advanced theory

of integration (Smith, p. 371).

References mentioned above -->

John Kelly, "General Topology", Van Nostrand, 1955.

Edward J. McShane, "Partial orderings and Moore-Smith limits".

In J. C. Abbot (editor), "The Chauvenet Papers: A Collection

of Prize-Winning Expository Papers in Mathematics", The Math.

Assoc. of America. [(Renfro) This is a reprint of McShane's

paper from the American Math. Monthly 59 (1952), 1-11.]

F. R. S. Temple, "100 Years of Mathematics: A Personal

Viewpoint", Springer-Verlag, 1981. [(Renfro) The form "George

Temple" seems to be more commonly used for his name.]

--------------------------------------------------------------

2. Fyodor A. Medvedev, "Scenes From the History of Real Functions",

translated by Roger Cooke, BirkhĂ¤user Verlag, 1991.

(top of p. 23) The concept of a limit is fundamental in topology.

Besides the fact that it arose and developed in analysis and the

theory of functions, it is recognized that the concept of a

generalized limit or the limit of a filter, which plays such

a fundamental role in topology, was born in the Shatunovskii

theory of real numbers and in the Moore theory of integration.

Shatunovskii introduced it in his lectures of 1906-1907, which

were published in lithograph form at that time. These lectures

were not printed until 1923 [3]. Moore made his first approach

to a generalized limit in 1915 in connection with the theory

of integration [1]. In a joint article with Smith [2] in 1922

this limit was introduced in general form. A similar concept

was introduced independently in 1923 by Picone. [fn-11]

Relevant footnote on p. 23 -->

(fn-11) We have not been able to consult the book of Picone

"Lezioni di Analisi Infinitesimale" (Catania, 1923).

(Renfro) Picone's book at least appears to exist -->

http://www.emis.de:80/cgi-bin/jfmen/MATH/JFM/?type=html&an=JFM%2049.0172.07

References mentioned above (numbering changed for present use) -->

[1] Edward H. Moore, "The definition of limit in general analysis",

Proc. National Acad. Sci. (U.S.A.), Ser. A, 1 (1915), 628-632.

[2] Edward H. Moore and H. L. Smith, "A general theory of limits",

Amer. J. Math. 44 (1922), 102-121.

[3] S. O. Shatunovskii, "Introduction to Analysis" (Russian),

Odessa, 1923. [(Renfro) I wasn't able to verify the existence

of this book. I searched at <http://www.emis.de/MATH/JFM/full.html>

using "Russian" for the language in one window, "book" for

Doc. type in another window, 1922-1925 for date restriction,

"100" for number of items displayed per page, and then I looked

at each item that my internet browser's 'find in page' option

gave me for the word "Odessa".]

--------------------------------------------------------------

3. Edward J. McShane, "Partial Orderings and Moore-Smith limits",

Amer. Math. Monthly 59 (1952), 1-11.

(footnote on p. 3) Kelley writes me that this [the term "net"]

was suggested by Norman Steenrod in a conversation between

Kelley, Steenrod and Paul Halmos. Kelley's own inclination

was to the name "way"; the analogue of a subsequence would

then be a "subway"! Since a stream with its tributaries is a

good example of a system directed by the relation "downstream

from," I incline toward "stream" rather than "net." But "net"

has the great advantage of having seen print first.

--------------------------------------------------------------

4. Robert G. Bartle, "Neetsw and filters in topology", Amer.

Math. Monthly 62 (1955), 551-557.

(from 1'st paragraph on p. 551) It appears that the former

notion [net] is predominant in this country, while the filter

theory reigns supreme in France.

--------------------------------------------------------------

5. Heinrich Reitberger, "The contributions of L. Vietoris and

H. Tietze to the foundations of General topology", pp. 31-40

of "Handbook of the History of General Topology", Volume 1,

edited by C. E. Aull and R. Lowen, Kluwer Academic Publishers,

1997. [Mentioned in Marc Olschok's post.]

http://informatik.uibk.ac.at/users/reitberg/Vietoris_Tietze.pdf

(2'nd paragraph on 1'st page) It is one of the purposes of this

article to show that L. Vietoris essentially carried out thie

program about 20 years before H. Cartan, in his fundamental work

during 1913-1919 on "connected set" (which was created mostly on

the battle field). In December 1919 he submitted this manuscript

as doctoral thesis to Escherich and Wirtinger in Vienna. In

1921 the version refereed by H. Hahn was published in the

"Monatshefte f. Math. u. Phys." [24].

(beginning of 2'nd Section, 2'nd page) ... but the priority

belongs to Vietoris who introduced this concept as "oriented

set" in [24, p. 184]. Interestingly enough, Birkhoff quotes

this paper concerning the separation axioms on p. 174 but doesn't

seem to have read any further!

Incidentally, Vietoris died less than a year ago (April 9, 2002).

He was 110 at the time (and, in fact, was less than two months

from turning 111) and was the oldest person in Austria when

he died.

--------------------------------------------------------------

6. My sci.math post from April 18, 2000

http://mathforum.org/discuss/sci.math/m/266055/266056

I give a fairly thorough list of references for the

use of nets/filters in real analysis.

--------------------------------------------------------------

Dave L. Renfro

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