# Talk:Gδ set

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Field:  Topology

## quesstion

Would it perhaps make more sense to combine the ${\displaystyle G_{\delta }}$ and ${\displaystyle F_{\sigma }}$ 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)
If the articles got merged we'd have an article titled "F-sigma and G-delta sets" that would cover both each type individually as well as their dual relationship. Although in some respects F-sigma and G-delta sets are completely different, I think a merge would be best considering the size of the F-sigma article: its extremely small and doesn't look like it'll be expanded any time soon. --Blacklemon67 (talk) 01:36, 21 April 2015 (UTC)
That's an awkward title, though. Ordinarily a WP article should talk about a single thing named by its (usually grammatically singular) title. There are exceptions, but only when fitting it into that paradigm is especially difficult for some reason.
In this case, I don't see any special difficulty. Sure, the F-sigma article is short. So what? Let it be short. --Trovatore (talk) 02:11, 21 April 2015 (UTC)
I hear what you're saying. For what it's worth, I made a demo merge here to see how well the two topics inhabit the same article. I found myself finding a lot of correspondences in the examples and basic properties. In the end nearly everything in the F-sigma article reduced to "likewise"es and "vice-versa"s in the G-delta article. --Blacklemon67 (talk) 06:34, 21 April 2015 (UTC)

## Continuous function

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 ${\displaystyle \Pi _{2}^{0}}$ formula - the formula states that for every natural number E > 0 there exists a natural number N > 0 such that whenever ${\displaystyle 0<|x-p|<1/N}$, we have ${\displaystyle |f(x)-f(p)|<1/E}$. 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

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)