In French mathematics, what is the difference between saying "l'ensemble est fermé" and "l'ensemble est clos"?
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4Welcome to French SE! Careful with the accents. In French, "ferme" and "fermé" are two different words with different meanings. I suppose you meant to say "l'ensemble est fermé" (as in the set is closed), instead of "l'ensemble est ferme" (as in the set is firm).– Thomas FrancoisCommented Jun 28, 2016 at 10:16
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1 Answer
The usual word is fermé in topology (« un fermé est le complémentaire d'un ouvert » = “a closed set is the complement of an open set”), and clos for closure under operations (« corps algébriquement clos » = “algebraically closed field”; “clôture algébrique” = “algebraic closure”).
While there is a connection between the two concepts — a closed set in topology is closed under the operation “taking a limit”¹ — the two adjectives are not interchangeable. There are even cases where the two words have different meaning: the fermeture algébrique of a subfield K in a field L is the set of elements of the field L that are algebraic over the subfield L, whereas the clôture algébrique of K is an algebraically closed superfield, which is different if L is not algebraically closed.
¹ Restrictions apply. Consult a mathematics text for details.
answered Jun 28, 2016 at 12:07
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Great answer. If "clôture" is an algebraic concept then I'm not sure in what context one would say "l'ensemble est clos". There has to be context to explain under what operations the ensemble is clos.– qobaCommented Jun 28, 2016 at 14:03
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The precise statement that has me baffled is: Si le groupe G est clos, les sous-groupes g fermé dans G sont les sous-groupes clos. Taking clos to mean algebraic closure seems in conflict with the description sous-groupes, which in modern usage already implies algebraic closure. It could be, however, that older definitions of a group omitted the requirement for algebraic closure. The text I am reading is from 1930. Commented Jun 29, 2016 at 11:45
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2@TulliusAgrippa Closure isn't necessarily algebraic closure (i.e. closure for taking roots of polynomials), it could be closure for any set of operations. Obviously it isn't the group operation here. The context should clarify what the operation is. This being said, it's possible that usage has changed since 1930 (which is pre-Bourbaki, Bourbaki caused some changes in algebra terminology). I've never studied older algebra texts so I wouldn't know. Commented Jun 29, 2016 at 21:51