Filters: Usage and History

20 views
Skip to first unread message

David R MacIver

unread,
Jan 23, 2003, 9:31:19 AM1/23/03
to
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?

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.)

Arturo Magidin

unread,
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?

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

G. A. Edgar

unread,
Jan 23, 2003, 11:56:38 AM1/23/03
to
In article <3E2FFCB7...@spoon.cam.ac.uk>, 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?

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/

Marc Olschok

unread,
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?

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

Dave L. Renfro

unread,
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

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, Birkhäuser 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

Reply all
Reply to author
Forward
0 new messages