User talk:Topology Expert/Archive 4

Conventions
Hello. Could you try to follow the usual conventions followed here? Notice this difference:
 * (n-1) dimensional
 * (n &minus; 1)-dimensional
 * (n &minus; 1)-dimensional

The n looks different in the two cases; the minus sign looks different (in particular, it looks different from the hyphen); and spaces precede and follow the minus sign. I've made the spaces "non-breakable", i.e. if the browser window is otherwise arranged so that a line-break would come between the n and the minus sign, that won't happen.

Note: variables are italicized; digits and punctuation (such as parentheses) are not. This matches the style that TeX uses.

Also, you don't need to write R^3; you can write R3.

With something like f^(-1)(x), you can write &fnof;&minus;1(x). The &fnof; looks different; the minus sign looks different; an actual superscript is used instead of a carat; the x is italic. I've put a non-breakable space before the minus sign since in superscripts on some browsers leaving no space there impairs legibility. Michael Hardy (talk) 16:47, 13 June 2008 (UTC)

(Replying to your comments on my talk page): I had in mind, NOT that you should use LaTeX (all of the notation in my paragraph above is non-TeX notation but rather I was writing about conventions followed in non-TeX notation as above. On Wikipedia, I prefer to use TeX only in "displayed" as opposed to "inline" settings, because on many browsers the sizes of the characters fail badly to match those of the surrounding characters, and fail to align correctly.  If you click on "edit" for this section of your talk page, you'll see how I created all of the notation above, and none of it uses TeX. Michael Hardy (talk) 18:15, 14 June 2008 (UTC)

Wikify - looks good

 * In reply to [ your message]:

Yup, you've mostly addressed the wikify tag, so it can be removed. There is still some wiki work to be done, but it is more minor now. The numbered list should be turned into prose (eventually), and some of the links are to WP:disambiguation pages, so need to be fixed. JackSchmidt (talk) 13:19, 15 June 2008 (UTC)

Locally finite group
Howdy, I cleaned up locally finite group and added references. I think the article is basically good now, though it is missing the work of Šunkov, Kegel, and Wehrfritz on the rank of locally finite groups. The part about formation theory could be expanded, but first wikipedia needs articles on the finite case. Some of the results can be strengthened, for instance to locally soluble, but again we do not have a wikipedia article on local solubility, so I think things are fine for now. Once wikipedia's group theory is more mature, one can add more history, but most of the big names in this area are still red links. JackSchmidt (talk) 16:13, 20 June 2008 (UTC)

Compact Hausdorff with no isolated points are uncountable
Howdy, I'm not sure what the objections to the proof were (I didn't see anything wrong, but I didn't look too hard), but I think I recall this as being a fairly standard proof and a reasonably common "finite intersection property is useful" example. Could you give a citation for the proof (I suspect it is in Munkres) for finite intersection property? Correct proofs are still WP:OR, but this proof I think is standard and easily cited.

I tried to adjust some minor things that might have irritated other editors about the proof, so I suspect it won't be reverted again in countable set. In F.I.P. you might still have trouble, but not if you give a citation. I think something similar is in Rudin's Principles, but instead of "not discrete" he assumes "connected".

BTW I think there is a topological proof of the infinitude of primes that is vaguely similar. It might have a home somewhere on wikipedia, though I'm not sure offhand where. JackSchmidt (talk) 21:39, 20 June 2008 (UTC)

Hausdorff Dimension of the Discrete Space
Dear Oded,

I wrote that a countable discrete space always has Hausdorff dimension 0 (whether it is countably infinite or finite). Since you said that this was incorrect, I am giving a proof of this statement. Could you please have a look at it? I am not going to worry too much about notation because I think you will understand what I mean.

Proof:

Let X be a finite discrete space having k elements. If p > 0, we will show that the p-dimensional Hausdorff content of X is equal to 0. This will prove that the infimum of the set of all positive real numbers, p, such that the p-dimensional Hausdorff content of X is 0, is 0. First of all, if e > 0, we will find an open cover of X by balls such that the sum of the pth power of their radii is less than e. Note that if we choose n > (k/e)^(1/p), then a cover of X by balls of radii 1/n will satisfy the desired property since:

n > (k/e)^(1/p)

n^p > (k/e)

e > k*(1/(n^p))

so that the sum of the pth power of their radii is less than e. Therefore, the p-dimensional Hausdorff content of X is 0 so that the Hausdorff dimension of X is 0 as desired.

Now suppose X is a countably infinite discrete space. We will show (once again) that if p > 0, the p-dimensional Hausdorff content of X is 0. Let e > 0 ; we note that there exists a sequence, (s_n), of positive real numbers less than 1 such that the sum of the pth power of each of their terms is less than e (which follows from elementary real analysis). If we consider balls indexed by the integers of radii (s_n), then the sum of the pth power of the radii will also be less than e. This means that the p-dimensional Hausdorff content of X is 0 since e was arbitrary. Therefore, the Hausdorff dimension of X is 0 as desired.

Q.E.D

I am quite sure that this proof is correct and therefore the Hausdorff dimension of a countable discrete space is 0. After having a look at the proof, please let me know if you agree. If you don't, then could you please point out any problems to me?

Thanks

Topology Expert (talk) 06:31, 24 June 2008 (UTC)


 * But you wrote in Hausdorff dimension that the dimension in that case was 1 and that this gives an example where the Hausdorff dimension and the Lebesgue covering dimensions were different. Why did you write that? Also, you have yet to explain to me why you edited the article on almost sure to say in effect that nowhere dense sets have zero measure. It seems that quite often you are "making up" mathematics and posting it on WP article, which is of course prohibited, as you know. Moreover, very often these statements turn out to be false. That is counterproductive. Oded (talk) 04:14, 25 June 2008 (UTC)