I am wondering why a standard notation for open sets is $G$ and that for closed sets is $F$. I mean, $F$ precedes $G$ in the alphabet, whereas open sets are usually introduced before closed ones.
$\begingroup$
$\endgroup$
12
asked Feb 10, 2023 at 17:31
-
3$\begingroup$ Fermé and geschlossen? (Why in different alphabets, I couldn't tell you, except that surely everyone is glad that we don't use $O$, tho' it could mean open or ouvert.) $\endgroup$– LSpiceCommented Feb 10, 2023 at 17:39
-
3$\begingroup$ Fermé sounds like a plausible explanation, but geschlossen means closed rather than open. $\endgroup$– Emil JeřábekCommented Feb 10, 2023 at 17:59
-
6$\begingroup$ If $G$ is supposed to be used for an open set, then 'Gebiet' would be my first association. $\endgroup$– Matthias KlupschCommented Feb 10, 2023 at 18:00
-
1$\begingroup$ Do you have a reference to a standard text where $G$ is used preferentially to denote an open set? $A$ is often used to denote a closed set because of the German 'abgeschlossen'. Similarly, $U$ for an open neighbourhood derives from the German 'Umgebung'. $\endgroup$– TyroneCommented Feb 11, 2023 at 7:19
-
3$\begingroup$ I think the association with F and G is through the notion of $F_\sigma$ and $G_\delta$ sets. $\endgroup$– Zhen LinCommented Feb 12, 2023 at 3:00
|
Show 7 more comments
1 Answer
$\begingroup$
$\endgroup$
11
I see that @LSpice has already provided an answer in their comment (I see that Emil has added a clarification, so to speak). Mine will compliment the comment by @LSpice a little.
Historically, closed sets were before the open sets (I believe so). Kazimierz Kuratowski defined topology (of general $T_1$-spaces} via the closure operation hence the closed sets came before open sets. We are talking here about the years 1921 and 1933.
I seem to remember that also Wacław Sierpiński was introducing the closed sets before open sets. Possibly, the respecive Sierpiński's script was the first monography entirely devoted to topological spaces. Sierpiński wrote also a separate script about metric spaces from the purely metic (not topological) point of view.
Earlier, I believe, that topologists/mathematicians were talking about neighborhoods rather than open sets. Perhaps(?) people started to talk about open sets seriously only after the results/papers by Paul S. Urysohn (I am not a historian -- please, do double-check my vague reminiscences).
Finally, the notation F (for closed sets) comes from the French "Fermé"; then open G followed. In the old days, French and German were ruling mathematics before English.
SOURCES:
- Hausdorff definition of topological spaces via neighborhoods:
$\qquad$https://mathworld.wolfram.com/HausdorffAxioms.html
See also Bourbaki foundational text on General Topology, it presents the Hausdorff neighborhood axioms soon after the today standard definition via open sets.
Hausdorff gutsy topological axioms absorbed Hilbert's axioms on 2d-geometric spaces.
-
1$\begingroup$ Thank you for politely passing over my suggestion that "geschlossen" might be a good inspiration for the name of an open set. $\endgroup$– LSpiceCommented Feb 10, 2023 at 19:23
-
1$\begingroup$ Even though you did compliment, I think you meant "complement". $\endgroup$ Commented Feb 10, 2023 at 23:44
-
1$\begingroup$ Thank you for your answer. If this is not too much to ask for, do you have sources for these interesting statements? $\endgroup$ Commented Feb 12, 2023 at 0:34
-
3$\begingroup$ Hausdorff's "Mengenlehre" is readily available st srchive.org. Starting at p. 209 he introduces neighbourhood systems axiomatically and goes on to use this to define interior points and open sets (which he calls "Gebiet") in the manner which was to become familiar. In this vein let me point out that the Austrian mathematician Vietoris extended this to develop the modern concepts of filter bases and nets, their convergence and compactness in the modern sense shortly afterwards (published 1919). $\endgroup$– terceiraCommented Feb 12, 2023 at 11:31
-
2$\begingroup$ @Iosif Pinelis: A useful reference is The emergence of open sets, closed sets, and limit points in analysis and topology by Gregory H. Moore (2008). On p. 229 (lines 2-3) Moore observes that Lebesgue, in his 1905 descriptive set theory paper, used F (fermé) for closed sets and O (ouvert) for open sets; and on p. 237 (first sentence of Section 15) Moore observes that Hausdorff used F (fermé) for closed sets and G (gebiet) for open sets. $\endgroup$ Commented Feb 12, 2023 at 16:10