Talk:Gδ set

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 Field: Topology

quesstion[edit]

Would it perhaps make more sense to combine the G_{\delta} and F_{\sigma} articles into one article on Borel sets? --68.102.149.76 20:40, 16 September 2006 (UTC)

Seconded. The notions are surely strictly equivalent. Richard Pinch (talk) 18:34, 17 June 2008 (UTC)
To be clear, Gδ and Fσ are not synonyms, but they are very closely related (a subset of a topological space is Gδ if and only if its complement is Fσ). The rationals are Fσ but not Gδ. Borel sets are more general than both, but again in a very closely related way. Since Fσ set is so short, I think it might be reasonable to just merge it into this article. Since Borel set has two contradictory meanings, it might be best to kee those articles separate, but mention that Fσδσδσδ… is another word for (the first kind of) Borel set. JackSchmidt (talk) 20:42, 20 June 2008 (UTC)
Hrm, I think Trovatore made a reasonable point at Fσ set. At least it made the merge less obviously a good idea. Again with Fσ being so short, it seems easy to merge. JackSchmidt (talk) 20:51, 20 June 2008 (UTC)

Continuous function[edit]

The fact that the set of points where aq function f is continuous is a Gδ set follows immediately from the fact that continuity at a point p can be defined by a \Pi^0_2 formula - the formula states that for every natural number E > 0 there exists a natural number N > 0 such that whenever 0 < |x-p|< 1/N, we have |f(x) - f(p)| < 1/E. If you fix a value of E, the set of x for which there is a corresponding N is an open set, and the universal quantifier on the E corresponds to the intersection of these sets. — Carl (CBM · talk) 12:15, 16 March 2008 (UTC)

Good point – I was wondering why this wasn’t in the article. I’ve added the above, wikifying some.
Thanks!
Nils von Barth (nbarth) (talk) 17:57, 31 August 2008 (UTC)
BTW, I believe that there is some converse (something like “Every Gδ can be realized as the points where some function is continuous”), but I forget the conditions – anyone know?
Nils von Barth (nbarth) (talk) 18:02, 31 August 2008 (UTC)

translation of Gebiet[edit]

Area is a misleading translation of Gebiet. I would say Gebiet means a geometric "domain", whereas "area" con be understood as the real number associated to it. —Preceding unsigned comment added by 78.129.56.66 (talk) 18:10, 9 January 2010 (UTC)

I agree that the translation of "domain" as "area" is misleading. From my experience, the German word "Gebiet" is usually used synonymously to the English word "neighborhood" in mathematics.-- Dr. scrubby-brush (talk) 01:13, 30 March 2010 (UTC)


Wait a minute — the article now claims that the G in Gδ is for Gebiet??? I thought it was for geöffnet. --Trovatore (talk) 21:20, 26 June 2010 (UTC)


The German word "Gebiet" means exactly what the English notion of "domain" means, namely an open connected set. A neighborhood is usually a domain (and therefore a Gebiet) but generally, one only speaks of "a neighborhood of some point or set", whereas a doman does not specify the location. I am not suer whether the G stands for Gebiet or geöffnet, but the translation of "Gebiet" neighborhood does not make much sense to me. Area might be possible, but as mentioned before, this is more likely to be the real number than the subset itself. Definitely "domain" is the best translation of Gebiet, see also the Wikipedia pages on Gebiet (Mathematik) and domain (mathematical analysis). As a reference, I am a German geometer who has studied both in the USA and in Germany, so I am familiar with both the German and the English terms. — Preceding unsigned comment added by Nemoline (talkcontribs) 21:34, 21 February 2013 (UTC)