799:. Let me remove from the Cantor space all strings that terminate with an infinite sequence of 1's. The result is a field of sets, I think: closed under complements, finite unions and finite intersections. There is no way to create a point with an infinite run of all-one's with only a finite number of intersections and unions. (right? or am I hallucinating?) The definition of "field of sets" says "closed under the intersection of pairs of sets", not "closed under countable intersections". So this construction meets the definition of a field of sets. The points of this field can be placed in one-to-one correspondence with the reals, because I removed the double-counting of the dyadic rationals. The natural topology on the reals has Lebesgue covering dimension of one, so I've exhibited a field of sets that is a proper subset of the Cantor set that has dimension one, not zero. So again, where did this dimension-zero thing come from?
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as much as possible of the subject to this simple model, before he is ready to see the subject in a thousand guises and models. While some concepts warrant separate introductory and advanced treatment, I believe most articles should have a reasonably pedagogical structure where the generailzations and alternative models are introduced toward the end. Readers at a higher level can skip sections 2 through n. They will probably discover that the articles have this structure.
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Mathematics is generally about abstractions and generalizations, and so even a concept like a
Boolean algebra is treated as an abstract concept. However, an encyclopedia article may be the wrong place to achieve the maximum abstraction and generality. The reader needs a simple model, and needs relate
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I am not implying that just fixing the above passus will fix the page. What is the deal with the Stone representation? Why is a representation needed? The whole treatment lacks all forms of motivational stuff. Making wikipedia's math pages a good source for half-educated readers is a monumental task.
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It is true that seeing cross connections between the various fields of mathematics is part of the fun, but reading wikepedia math articles has become an almost impossible task because every link one follows trying to get a foundation of the terminology used in a first page, creates the need to follow
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The page does contain links already, and perusing each does allow the reader to dechiffer the text. However, I would like to encourage contributors to make this task less ardous, even if that means repeating some stuff in a number of pages. There could also be a link that is singled out early in the
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So what happened? I guess we are now talking about the same field of sets but using a different topology? Or perhaps we threw away some of the complexes along the way, so that its a different "field of sets", but built on the same set X? ?? By the way, does it matter when X is countable, vs. when
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A glaring problem in the very first sentences is a failure to distinguish finite vs countable number of operations. This propagates to the section on topology: in a topology, one is allowed countable unions, finite intersections, and there is no talk of complements, because that would ruin things
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The statement makes perfect sense once the general mind frame is clear, but it is not very helpfull to the reader that lacks this frame. The sheer number of questions that arise in a reader's mind makes it hopeless to even start guessing. Perhaps part of the problem is a mismatch between the pages
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to the set of atoms). I also wrote intersections as consequences, not definition, both in the introduction and in the sigma algebra section. For example, Rudin's Real
Analysis defines sigma algebras by requiring complements and countable unions, mentioning intersections as a consequence (by the
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Yes, we are talking about the same field of sets but using different topologies. This is usual enough. And I do not agree that "measurable spaces are typically not zero-dimensional". There are a number of (not always equivalent) definitions of dimension for topological space, but no one defines
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but uses the empty set and even formally requires intersections, so it is not a good reference for the definition section. I added it to "External links" anyway. Preferably there would also be a new section "Definition" that would be very clear and precise.
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There isn't anything wrong in the article, the problem is that you do not understand what the term "base for a topology" means. The article does link the
Knowledge article explaining it, I don't think we need to explain standard topology terms again in this
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and insert some additional sets into the complex of the field of sets? Somehow, this is all very confusing. I'm looking for precise, exact definitions, and they're not here. I'm not even vaguely expert enough to fix this without adding additional errors.
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I suppose that in the first years it has been more beneficial for the public that contributors quickly wrote about many subjects, but we need that the structure be cleaned up eventually, and I am hoping to give another push in the right direction.
239:(5) What is a join? I came to this article searching for a definition of "an algebra", and most other wiki pages I found seemed wrong since they speak mostly about bits and bit strings - not to mention the pages about 'algebra' in general.
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The article does not state that a topological space that also happens to be a measureable space, is zero-dimensional. It says the topology generated by the complexes (measurable sets) is zero-dimensional. The only one confused is
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What do you think? In this wording, I am trying to reduce the number of underlying concepts the reader needs to know about. Also, I try to make it optional to bother about the order theory and atoms, while still offering that
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to stop skimming articles and coming to knee-jerk reactions. Still, the lack of discussion of cardinality is vaguely un-nerving, as this is usually important in many other areas, and so seemingly should matter here as well.
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for a topology (the complement of an open set is not open; adding complements gets you Borel sets, instead, which are something different). So there is some kind of creeping incorrectness hard at work in the article :-(
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and this page. I suggest we try to keep most pages as self-contained as possible, and include enough words of introduction of the concepts used, or, when available on wikipedia, links.
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It's also missing the basic infinite examples (before one gets to extra structure) like finite-cofinite or intervals, which one can even find in
Springer EOM or in Givant and Halmos
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I corrected the incorrect definition: X must be an element of F, that is, an algebra F' over a strict subset Y of X is not define an algebra over X (unless we add
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I just added an "expert attention" tag to the article. Pretty much every issue raised in all of the different comments above needs to be addressed and resolved.
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could be replaced by the formally (but not truly strictly) weaker "F is nonempty", as X equals the union of any complex with its complement. But I still prefer
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three more links to understand the terminology in the new page. I hope contributors will eventually find time to take most articles down to a simpler level.
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Some passages need rewording to become intelligible for an audience that does not already know. At the moment I have in mind the statements
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it isn't? Boolean algebras for finite, countable and uncountable sets are rather different from one-another. What happens if I do
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Began the painful
Texification process, bear with me. Lot more to say in this article but not finding as much time as I'd like.
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I thought I understood what a base was. I don't understand how we got to "zero-dimension". Is zero-dimension referring to the
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This article is deeply confusing. Aside from the multiple unanswered, orphaned comments above, I also see stuff like this:
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That is certainly NOT what it says! It says that ALL fields of sets are zero-dimensional. It doesn't say "some of them".
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Said all I have to say for now, if anyone knows of any other interesting uses or aspects of fields of sets, please add.
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A clarification just added to the article: "...just one of notable topologies on the given set of points..."
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as a totally disconnected set? In that case, I guess I agree: if I think of only the points, the atoms of
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endowed with the box topology, which would make it zero-dimensional. But certainly, there are subsets of
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until a consensus is reached, and readers of this page are welcome to contribute to the discussion.
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Every finite
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One obvious question is if there are finite fields of sets other than the powerset etc.
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Isn't the "subset of the powerset of X" just a 'subset of X'? 14:51, 10 Jul 2007 (UTC)
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Made an own page for this as fields of sets are much wider than just sigma algebras.
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Have been busy on other pages but will add more to this page soon. - 13 Oct 2004
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Enquiring minds want to know. Because later on, this very same article states:
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Currently too much ugly formal notation, will try rewording to avoid notation.
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page as the one leading to an exposition of the terminology used in the page.
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Great! Except that measurable spaces are typically not zero-dimensional.
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I'm curious because Mac Lane and
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Well, I just rewrote this paragraph; hopefully, more accessible now.
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such that the same boolean algebra can be represented by the set of
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for a topology, we denote the corresponding topological space by
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139:on the
1186:Rigmat
1112:Rigmat
821:unions
242:under
36:scale.
959:Yes.
436:etc.
400:linas
380:linas
293:atoms
263:, if
229:whole
221:atoms
1229:talk
1190:talk
1116:talk
965:talk
951:talk
937:talk
923:talk
896:talk
878:talk
847:you.
506:base
442:talk
424:talk
404:talk
384:talk
362:talk
335:talk
320:talk
915:all
911:all
395:got
295:of
277:all
200:No
131:Low
1243::
1231:)
1192:)
1165:∈
1135:∈
1118:)
1110:--
1090:∈
1056:∈
1053:∅
1026:∈
967:)
953:)
939:)
925:)
898:)
880:)
587:?
492:⟩
476:⟨
444:)
426:)
406:)
386:)
364:)
356:.
337:)
322:)
299:.
1227:(
1188:(
1170:F
1162:X
1140:F
1132:X
1114:(
1095:F
1087:X
1061:F
1031:F
1023:X
996:c
992:Y
963:(
949:(
935:(
921:(
894:(
876:(
835:.
782:N
778:=
775:X
740:X
736:2
713:X
709:2
684:X
662:X
658:2
637:X
617:X
574:.
558:)
554:X
550:(
547:T
527:)
523:X
519:(
516:T
487:F
482:,
479:X
473:=
469:X
440:(
422:(
402:(
382:(
360:(
333:(
318:(
297:A
289:Y
281:Y
273:X
269:Y
265:A
261:X
257:A
143:.
42::
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