Talk:Fσ set

Merge into G-delta?

Well, I'll say it if no-one else will ... should this article be merged with G-delta set? Richard Pinch (talk) 18:29, 17 June 2008 (UTC)

Hmm--not implausible, but my immediate reaction is probably not. Theoretically they could both be merged into Borel hierarchy, but I think it's probably useful to have separate articles at this very low level of the hierarchy, as there are significant properties that can be mentioned that are not the same (for example, a G-delta subset of a Polish space is itself Polish; I don't know of any dual result for F-sigmas). --Trovatore (talk) 18:34, 17 June 2008 (UTC)

I would agree with the merge, but Trovatore's point is well made. JackSchmidt (talk) 20:44, 20 June 2008 (UTC)

The terminology

The terminology definitively goes back to Felix Hausdorff, 1914:

Felix Hausdorff, Grundz\"uge der Mengenlehre, 1914 (Reprinted Chelsea New York 1949; Preface: Greifswald, March 15, 1914.) viii+476.

P. 23: \S 10 $\sigma$-Systeme und $\delta$-Systeme. Here the author defines a $\sigma$ System to be a set of subsets of a set which is closed under the formation of countable unions, and the concept of a $\delta$-System is defined dually.

P. 215: Here the author defines a subset of what later is called a ``topological space (defined here in terms of (open) neighborhoods) to be a {\it Gebiet} if it is open. Note the {\bf G}.

P. 305: Verbal citation: Wir bezeichnen abgeschlossene Mengen, wie schon \"ofter mit $F$ (ensemble ferm\'e) (sic), Gebiete mit $G$. Die $F_\delta$ sind wieder abgeschlossen, da ja sogar der Durchschnitt beliebig vieler abgeschlossener Mengen wieder abgeschlossen ist; dagegen sind die $F_\sigma$, zu denen u.~a.\ die abz\"ahlbaren Mengen geh\"oren, im allgemeinen nicht abgeschlossen. Ebenso sind die $G_\sigma$ wieder Gebiete, die $G_\delta$ nicht. Die $F_\sigma$ und $G_\delta$ sind n\"achst den $F$ und $G$ selber die einfachsten Punktmengen, danach folgen die $F_{\alpha\delta}$ und $G_{\delta\sigma}$ usw. Wir wollen alle diese Mengen als {\it Borelsche Mengen} bezeichnen. $\dots$ End verbal citation.

Translation:

 We denote closed sets, as we did frequently, by $F$

(ensemble ferm\'e), open sets by $G$. The $F_\delta$ are closed since the intersection of even arbitrarily many closed sets is closed again; by contrast, the $F_\sigma$, to which the countable sets belong, are not closed in general. Likewise, the $G_sigma$ are open again while the $G_\delta$ are not. The $F_sigma$ and $G_\delta$ are, right after the $F$ and $G$, the simplest point sets, followed by the $F_{\alpha\delta}$ and $G_{\delta\sigma}$ etc. We are going to call all of these sets {\it Borel sets}. $\dots$ End translation. 130.83.2.27 (talk) 13:03, 1 December 2010 (UTC)