Talk:Closure (mathematics)

Hi, I speak spanish, so forgive my english. This article is not linked with spanish version. the spanish version is "Ley de composición interna". I don't know how to link the pages. Thanks

This request is over two years old! Sigh.....Added.----occono (talk) 02:18, 22 August 2009 (UTC)

I think...

I think that the notion of "closure without qualifier", i.e. Closure (topology), referring to closed sets, should be made a little more visible. (Maybe some kind of "disambig list" at the end of the introduction.)

Also, I find that this article is written in a way which is somehow unnecessarily complicated... — MFH:Talk 22:12, 21 March 2006 (UTC)

Are you referring to the rewrite I did a few days ago? What about it do you not like? My main goal was to make it so that this page describes both the property called closure (a set satisfies this property if it is closed), as well as the closure operator (which maps each set to a closed set). If you have some specific complaints, I could try to address them. -lethe ^{talk +} 13:55, 24 March 2006 (UTC)

naturals not closed under subtraction

Natural numbers are not closed in subraction because one (natural number) minus (natural number) is zero (not a natural number). —The preceding unsigned comment was added by Ieopo (talk • contribs) .

You're right about that. I'm glad you agree with the article, which says in the first sentence that "For example, [...] the natural numbers are not [closed under subtraction]". Thank you for your help. -lethe ^{talk +} 19:11, 11 April 2006 (UTC)

Closed sets

The current version of the article states that.

An operation of a different sort is that of taking the limit of a sequence. A set that is closed under this operation is usually just referred to as a closed set in the context of topology.

This is not true. In a topological space a set is closed if and only if it is closed under taking the limits of nets or filters. Limits of sequences aren't sufficient in general. I would also say, that the article should refer to closure operator in the part about abstract closure operators. --Kompik 11:31, 12 April 2006 (UTC)

I agree with your points and have attempted to address them in the article. How do you like it now? -lethe ^{talk +} 14:46, 12 April 2006 (UTC)

The improvements appear to have been deleted, now. The uncorrect statement about closure in topology is still here.--78.15.196.42 (talk) 15:24, 25 February 2012 (UTC)

What about closure and repeating members?

If a set were to be {0, 1}, would that be closed under addition? 0+1 = 0, ok, 0+0=0, ok, but 1+1=2. Does this mean that it is not closed under addition? All the examples I can find use natural numbers or integers as sets. These don't matter for addition or division. —Preceding unsigned comment added by Nswartz (talk • contribs) 15:05, 16 August 2008 (UTC)

As you point out, 1+1 = 2 ∉ {0,1}, so, no, our set is not closed under the usual addition operation for integers. In fact, no finite, nonempty set of integers other than {0} is closed under integer addition: let S be such a set; by assumption, S contains some nonzero member e. Then e + e + . . . + e = ne ∉ S for some integer n, since e ≠ e + e and S is finite.

On the other hand, define an operation + on our set {0,1} by

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 0

("addition mod 2"). In this case, the answer is "yes". For closure properties of sets of integers under addition more generally, the entries on modular arithmetic and cyclic groups seem like reasonable places to start.

75.184.118.88 (talk) 16:33, 16 August 2010 (UTC)

Closure Properties

Closure property is a property which a set either has or lacks with respect to a given operation.A set is closednwith respect to the operation if the operation can always be completed with elements in the set . —Preceding unsigned comment added by 69.111.182.182 (talk) 01:54, 4 February 2010 (UTC)

Closures in Functional Programming Languages

A discussion of how this relates (or doesn't) to closures in functional programming languages might be useful. I don't consider myself qualified to discuss this yet. Although the concept of mathematical closure has been brought up in some lectures on Lisp I've seen, the closures in the language seem like a more concrete, perhaps separate concept from mathematical closure... but I could be wrong. In other words, is "closure" in functional programming overloaded to mean something else, or not? —Preceding unsigned comment added by 98.207.0.180 (talk) 17:55, 26 February 2010 (UTC)