# 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
See also math.stackexchange.com/a/404776/589. –  lhf Oct 9 '13 at 16:24
What is the meaning of "robust URL"? I spent several minutes googling all sorts of variations of this phrase without finding a definition or a context that was sufficiently jargon-free to guess at a definition. –  Dave L. Renfro Oct 9 '13 at 16:28
@DaveL.Renfro, I meant, one that goes directly to a given page on a given book, instead of just copying one that worked, and which probably contains session data and other junk. I'm concerned that the URLs I've posted may not work in the future. –  lhf Oct 9 '13 at 16:30
Perhaps this, from a not-yet-submitted paper I wrote a few years ago? References Note: In the references that follow, items in Google's digitized book archive are identified by a code that begins with "books?id=" (e.g. "books?id=uYdJAAAAYAAJ"). To obtain a URL for such an item, attach this code to the end of " books.google.com ". Also, to obtain a URL taking you to a specific page in such an item, attach the string "&pg=PA#" to the end of the full URL, where # is the string of decimal digits for the number of the page you want. –  Dave L. Renfro Oct 9 '13 at 16:38