# Why do we traditionally use letter U for open sets?

Most of traditional usages of symbols in mathematics have origin in English, German or French words that start with that letter, for an example: $p$ for a prime number, $\mathbb{Z}$ for integers (ger. Zahlen), $G$ for a group, $K$ for a field (ger. Körper) etc. Why do we use $U$ instead of obvious $O$ (open, öffnen, ouvert...)?

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I think it's $U$ for Umgebung (neighbourhood). –  Daniel Fischer Oct 9 '13 at 15:38
Note we also have $u$ for unit in some cases, or for $u$-substitution. I'm guessing that $O$ would be avoided on principle to limit confusion with $0$. –  abiessu Oct 9 '13 at 15:40
@DanielFischer, any reference? Cantor, perhaps? –  lhf Oct 9 '13 at 15:54
Btw, the German word for the adjective “open” is “offen”, “öffnen” is the verb. –  Carsten Schultz Oct 9 '13 at 17:01
@lhf If I had a reference I'd have an answer. –  Daniel Fischer Oct 9 '13 at 17:08

Daniel's suggestion that $U$ comes from Umgebung, which is neighbourhood in German, makes sense because several important German mathematicians are linked to topology. Among them are Cantor and Hausdorff.

The definitive reference is probably the book History of Topology edited by I.M. James. There, on page 213, we find a reference to notes by Hausdorff that mention Umgebung.

The book Foundations of Abstract Analysis by J. H. Dshalalow says explicitly on page 175 that $U$ comes from Umgebung.

Cantor seems to have been the first to use Umgebung (probably in 1872) but I couldn't find a good reference.

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Does anyone know how to find robust URLs for precise pages in Google Books? –  lhf Oct 9 '13 at 16:15