# Terminology: Lost in translation or multiple-meanings

I was reading Uniformization of Riemann Surfaces by Henri Paul de Saint Gervais (not a real person, but a group of French mathematicians), and the translator kindly points out that the name of "the greatest theorem of the 19th century" comes from the French word uniforme, meaning single-valued (as opposed to multi-valued). It may still seem obscure, but in the text (the introduction) it was explained pretty well.

For a lesser example, in the same introduction it talks about a group $\Gamma$ acting on the upper half plane (fixed point) freely and properly. That casual parenthetical remark cleared up the meaning of "free action" that I could never make sense of and had to look up its definition repeatedly.

I hope it is okay to ask for more examples of this sort. (You are welcome to rephrase the question.) It may not strictly be the original intention, but may have contributed to its wide acceptance but has since been forgotten (e.g. What is the naming reason of poles in complex analysis?). As other Terminology questions and answers show (especially in algebra), this may be quite tortuous to explain properly, and may not be more helpful than "just learn the definition."

For starters: what is so proper about a proper map?

Updated: To avoid being too vague that any terminology can have a story behind it (meromorphic, homology, etc), here are some guiding criteria:

1) Due to translation/importing (most often from French and German), or the multitude of meanings of the English word itself, the original meaning of the terminology has been lost in most texts on the subject;

2) it has been generalized out of the original context;

3) and that the original meaning helps in the understanding of the terminology, without having to give a long explanation.

• I recall reading that Grothendieck introduced the terminology "nuclear space" into functional analysis to call attention to the development of nuclear arsenals during the Cold War. I'm not sure if this is actually true (and I can't substantiate it at the moment) but it suggests that some terminology may not be rooted in its underlying meaning. Aug 5 '17 at 13:17
• It may well be true, but don't "nuclear spaces" satisfy a théorème des noyaux ? This seems enough motivation for the name... Aug 5 '17 at 15:32
• Why don't you make the assumption that "nuclear" has been chosen because of its double sense?
– YCor
Aug 5 '17 at 17:37
• I'm not sold in this being the origin of the terminology free action. The action is free in the categorical sense : if T is a transversal to the orbits then any map from T to a G-set extends uniquely. Aug 5 '17 at 18:27
• This is rather embarrassing, but it was only recently that I realized the term "quiver" (in the sense of directed graph) was coined because it refers to a collection of "arrows" (directed edges) Aug 16 '17 at 13:31