Talk:Cantor space

Space or Set?

What about 2^S where S is uncountable? Should that be included among what are called Cantor spaces? (It can be shown that every Boolean space, i.e., every totally disconnected compact Hausdorff space, is a compact subspace of one of these.) Michael Hardy 03:29, 1 Nov 2003 (UTC)

I was wondering the same thing. And this may seem unnecessarily fussy to remark, but if Cantor space is unique up to homeomorphism, why does this article keep referring to it with the indefinite article? I understand that there are many distinct, homeomorphic realisations of the Cantor set (space), but when talking about it where it doesn't matter what the concrete representation of it is, can't it just be called "the Cantor space"? We talk about "the long line", or "the Sierpinski space", not "a long line" or "a Sierpinski space". Revolver

If someone has a reference for referring to an uncountable product of 2, as a Cantor space, fine, though I've never heard that usage. But it's certainly not "Cantor space"; Cantor space is separable. --Trovatore 06:41, 18 February 2006 (UTC)[reply]

Willard's General Topology (p. 121) defines a Cantor space as any product (possibly uncountable) of a finite discrete space. The Cantor set is homeomorphic to 2^N and therefore a Cantor space. -- Fropuff 07:01, 18 February 2006 (UTC)[reply]

OK, sounds like something should be said about this usage, then. Whoever incorporates it, please make sure to distinguish a Cantor space from Cantor space. (The problem with the edit I reverted is that, to me and I think to most set theorists, "Cantor space" with no article means the unique topological space homeomorphic to 2^ω. Could be either "Cantor space" or "the Cantor space"; means the same thing. "A Cantor space" sounds different.) --Trovatore 07:11, 18 February 2006 (UTC)[reply]

We could remove the ambiguity by always calling 2^ω the Cantor set, and use Cantor space for the general case; with a remark to the effect that the Cantor space always refers to the Cantor set. (Err, maybe that's too confusing.) -- Fropuff 07:29, 18 February 2006 (UTC)[reply]

No, I don't agree with that. Descriptive set theorists call 2^ω "Cantor space", not "the Cantor set". The latter sounds like the middle-thirds set to me. I think we should keep 2^ω as the primary meaning of "Cantor space" with no article, and mention that some general topologists call 2 to any discrete space "a Cantor space". (I suspect the latter usage is not all that widespread; if I'm wrong about that we might have to do something like Baire space vs Baire space (set theory), except this time 2^ω should get primary billing.) --Trovatore 16:35, 18 February 2006 (UTC)[reply]

I'm with Fropuff on this, everything I've seen refers to 2^ω as the Cantor set; and Willard (opcit) seems to provide a definition of a Cantor space as something more general than what is described in this article. linas (talk) 16:43, 21 June 2010 (UTC)[reply]

I guess you mean it has a countable base because actually 2^R is separable and not Cantor space. Dskloet (talk) 07:49, 28 July 2009 (UTC)[reply]

Uhhh, well, I suppose that depends on what topology you put on 2^R. I had in mind just the product topology; that's obviously not separable, but of course it also makes no use of the topology of R. What topology do you have in mind? --Trovatore (talk) 10:09, 28 July 2009 (UTC)[reply]

Yes, I was thinking about the product topology as well. The collection of finite unions of rational intervals is both countable and dense, isn't it? (seeing a subset of R as an element of 2^R) Dskloet (talk) 21:41, 5 August 2009 (UTC)[reply]

I hadn't even seen this talk page when I created the article Cantor cube! Well, see it for arbitrary products of 2. Melchoir 21:14, 6 July 2006 (UTC)[reply]

Terminology

I seriously question the term "Cantor space" being used to mean -- as it is defined here -- a topological space homeomorphic to the Cantor set.

In my 40 years of being a topologist, I have never encountered a space homeomorphic to the Cantor set being called anything but a "Cantor set". (The original, traditional Cantor set is called the "middle-thirds Cantor set".)

I strongly urge this article to bring itself in line with prevailing terminology.Daqu (talk) 07:18, 26 May 2008 (UTC)[reply]

It is, in fact, called Cantor space in descriptive set theory. Try for example this search in Google Scholar. My sense is that a "Cantor set" is most likely thought of as a subset of R (or maybe Rⁿ), whereas the canonical realization of the Cantor space is the collection of ω-sequences of zeroes and ones, but obviously this is inessential, just my observation of a tendency in usage. --Trovatore (talk) 07:29, 26 May 2008 (UTC)[reply]

Thank you, Trovatore, for clarifying that (which also explains why I've never heard that usage as a topologist).

It remains also true, however, that in topology any space homeomorphic to "the" Cantor set is called "a" Cantor set (and this usage of the definite vs. indefinite articles pervades topology).Daqu (talk) 05:53, 31 May 2008 (UTC)[reply]

If the article is wrong about the terminology in topology, maybe we should put it into line with the "Cantor space" terminology in descriptive set theory, which Trovatore or I would have no difficulty sourcing. — Carl (CBM · talk) 23:15, 5 August 2009 (UTC)[reply]

FWIW, Cantor's original article is reprinted in a book "classics on fractals" which is fairly easy to obtain. He describes a "perfect set" first, i.e. the general case, and mentions the middle-third example only in passing, at the end, as a way of constructing a monotonic, continuous function whose derivative is zero on a set of measure 1. (I've never heard ot the term "Cantor space" before). linas (talk) 14:56, 21 June 2010 (UTC)[reply]