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Am I right in assuming that the word closure is derived from the word closed?

If so, I would be interested to know the name of this procedure and what it yields when applied to the word open.

My motivation is a mathematical one: "The closure of a set is the smallest closed set containing it." For certain topological spaces (Alexandrov spaces) it is true that every set is contained in a smallest open set.

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For the sake of the next sentence, let me use the word "opening." Can you give an example of a mathematical statement that you would like to express about the "opening" of a set? –  Robin Kothari Nov 2 '11 at 14:54
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Be more specific in your example. My topology is really rusty, but closed and open are not antonyms (they don't share the context), so it is possible for a set to be clopen. –  Unreason Nov 2 '11 at 15:04
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To help out answerers, the concept the OP is looking for is a word for the process of making something 'open'. 'Closure' is the process of making something 'closed' (e.g. 'take the closure of a set' gives you another set that is closed). So the word being sought is a process to make something have the property 'open'. –  Mitch Nov 2 '11 at 15:20
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@RobinKothari: That would be "The opening of a set is the smallest open set containing it." –  Rasmus Nov 2 '11 at 15:21
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3 Answers

My maths isn't good enough for me to grasp exactly what the closure of a set actually means. In fact, I can't even tell whether it's meaningful to speak of an "opposite" in this context.

By way of example, in perspective drawing you can speak of a vanishing point, but I don't think anything could meaningfully be called an "appearing point", regardless of the actual term used.

At the more general linguistic level, closure is effectively an alternative noun to the more common closing, though it's obviously acquired specialised meanings in the contexts of mathematical set theory and psychology/counselling. I don't think there's a corresponding alternative form for the "opposite" noun opening.

Per my answer to this question, I call the process by which we create words such as closure from close linguistic production. I don't know if the -ure suffix is still "productive" (ie - can be used to make "new" word-forms). I suspect not, but maybe it is used for obscure new technical terms.

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Thank you for your answer. Your third paragraph seems to answer my question. The first and the second, frankly, don't help answering the question and seem off-topic to me. –  Rasmus Nov 2 '11 at 15:25
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Well presumably you're the mathematician, so you would know if it's mathematically meaningful to conceive of something that's the "opposite" of the "closure of a set". I don't, so the first two paragraphs are just my way of saying that, and inviting anyone who knows more about it to enlighten me. But I am reminded I never addressed your exact question "what is this lingiustic process called", so I will edit for that. –  FumbleFingers Nov 2 '11 at 17:16
    
Thank you very much for your additions. –  Rasmus Nov 2 '11 at 17:41
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Not exactly. From EtymOnline

closure late 14c., "a barrier, a fence," from O.Fr. closure "enclosure; that which encloses, fastening, hedge, wall, fence," also closture "barrier, division; enclosure, hedge, fence, wall" (12c., Mod.Fr. clôture), from L. clausura "lock, fortress, a closing" (cf. It. chiusura), from pp. stem of claudere "to close" (see close (v.)). Sense of "act of closing, bringing to a close" is from early 15c. Sense of "tendency to create ordered and satisfying wholes" is 1924, from Gestalt psychology.

Although ultimately deriving from close, this word was not derived from it in English, but rather came directly from French. Whether it's some standard process or not is beyond my knowledge.

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In English in general, the noun form of open is openness, while the noun form of close is closure.

In math, the closure of a set is the smallest closed set containing it. The natural mathematical antonym of closure is interior; the interior of a set is the largest open set it contains.

But what you are looking for is a word for the smallest open set containing a set. For most families of sets, this isn't a useful concept, since such a set doesn't generally exist. I don't believe there is a pre-existing word for it in the mathematical literature. In my opinion, neither of the words opening or openness is a good term for this concept. You could conscript a new word to mathematical use, and call it the purview, or something similar. Or you could call it the least (or smallest) open neighborhood of the set.

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Concerning your first sentence: Isn't there also closedness? –  Rasmus Nov 2 '11 at 16:52
    
@Rasmus: Googling "closedness", I find that it is much, much rarer than "openness", and that nearly all the hits are math papers talking about open and closed sets. Outside of math, I don't know whether it counts as a word. It's not in Merriam-Webster's online dictionary. –  Peter Shor Nov 2 '11 at 17:31
    
That's interesting. I didn't know that closedness is not a word. –  Rasmus Nov 2 '11 at 17:41
    
@Rasmus: I would say closedness is a word, but one found only in the context of mathematics (like abelian). –  Peter Shor Nov 3 '11 at 0:38
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