User talk:Kompik

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1 Welcome!
2 Hi
3 PlanetMath
4 Limits and AC
5 Please vote on list of lists, a featured list candidate
6 Compass and straightedge

[edit] Welcome!

Hello Kompik, and Welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are some good places to get you started!

I hope you enjoy editing here and being a Wikipedian! Please be sure to sign your name on talk pages by using four tildes (~~~~) to produce your name and the current date, or just three tildes (~~~) to produce your name only. If you have any questions, or are worried/confused about anything at all, please either visit the help desk, or leave a new message on my talk page at any time. Happy editing, good luck, and remember: Be Bold!

FireFox^{T C E} 17:35, 5 September 2005 (UTC)

[edit] Hi

Hi Kompik. Thank you for your fixes to math articles. You are welcome to join the math club, at Wikipedia:WikiProject Mathematics (see its talk page, and the link to participants subarticle).

[edit] PlanetMath

Hi Kompik. Thank you for copying things from PlanetMath. One remark. When you mark an article as copied, removed the word "guess" from "WP guess" but do keep the word "WP". Also, put in the status. If you copied an article the status should be "C", if our existing article is adequate, it should be "A". See the table at the top of each page in the project for the other meanings. I fixed your entries at Wikipedia:WikiProject Mathematics/PlanetMath Exchange/11-XX Number theory. Cheers, Oleg Alexandrov 15:56, 25 September 2005 (UTC)

[edit] Limits and AC

Could you please explain your edit to ~~limit (mathematics)~~ continuous function where you say that "It's interesting that the proof of the equivalence of Cauchy's and Heine's definition of the continuity needs some form of the axiom of choice"? In fact, I can't see the difference between Cauchy's and Heine's definition. By the way, we usually write in a formal style, and thus we don't use contractions like "it's" but write them in full as "it is"; I changed this in the ~~limit (mathematics)~~ continuous function article. -- Jitse Niesen (talk) 21:44, 28 September 2005 (UTC)

If Kompik is thinking about the same thing as me, then it goes as follows. Assume that a function f is sequentially continuous at a point x. Need to prove that if B_n is an ever-shrinking sequence of open balls at x, then the values of the function f on the ball B_n all approach the value f(x_0). Assume the opposite. Then, you are forced to pick an element x_n from each ball B_n, such that f(x_n) does not go to f(x). This choice of x_n from each ball is the axiom of choice applied to the sequence of sets B_n. It is a weak form of the axiom of choice, because the axiom of choice works for an arbitrary family of sets, not necessarily a countable one. Oleg Alexandrov 00:54, 29 September 2005 (UTC)

Sorry about messing up the articles in my original question: I should have refered to continuous function instead of limit (mathematics). I understand what Oleg is saying, and I thought that Kompik was talking about something like that (but I couldn't remember the details). However, continuous function decribes Heine's definition of continuity as "The limit of f(x) as x approaches c must exist and be equal to f(c)." It's not clear what definition of limit is being refered to; in fact, the article limit (mathematics) does not seem to have the sequential definition, which goes something like: the limit of f(x) as x approaches c is L if, for every sequence x_n with limit c, the sequence f(x_n) has limit L (where the limit of sequences is as defined in Limit (mathematics)#Limit of a sequence). If I now understand it correctly, it seems some clarification is in order. -- Jitse Niesen (talk) 12:09, 29 September 2005 (UTC)

You are correct. It was my fault - I placed the noted that "this definition is Heine's" into continuous function. I had to read the entry more thoroughly - but at least that definition is similar to the sequential continuity - which I had in mind and which Oleg explained above. (See e.g. [1] for what I call Heine and Cauchy definition of continuity - but I think it's clear enoguh by now.) The solution is either to remove both references to Heine or to add Heine's definition somewhere. I'm rather busy these days. If some of you guys has time to do the corrections, please do it. Otherwise, I'll try to update the page later - I'm travelling abroad this weekend and I have some stuff to do before. Thanks a lot for pointing out the mistake. --Kompik 17:07, 29 September 2005 (UTC)

I've tried to fix it. Check the article, if you want. Probably you'll have some corrections. --Kompik 07:09, 30 September 2005 (UTC)

No need for corrections, but I did rewrite it slightly. -- Jitse Niesen (talk) 12:43, 30 September 2005 (UTC)

Finaly, I've corrected it to the following (it should be correct now) --Kompik 09:59, 3 November 2005 (UTC)

Cauchy's and Heine's definition of continuity are equivalent. The usual (easier) proof makes use of axiom of choice, but in the case of real functions it was proved by Wacław Sierpiński that axiom of choice is not actually needed.

One more note: The Sierpiński's result speaks only about global continuity. Maybe this is not important enough to mention it in the expository article, but I mentioned it -- to avoid confusion. --Kompik 14:30, 21 January 2006 (UTC)