Various concepts of "closure" or "completion" in mathematics - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T12:34:25Z http://mathoverflow.net/feeds/question/14509 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14509/various-concepts-of-closure-or-completion-in-mathematics Various concepts of "closure" or "completion" in mathematics Zev Chonoles 2010-02-07T16:56:15Z 2010-02-25T08:23:05Z <p>Out of idle curiosity, I'm wondering about all the various idempotent constructions we have in mathematics (they seem to be generally referred to as a "closure" or "completion"), and how some of them are related (e.g., the radical of an ideal and the closure of a subset of $k^n$ in the Zariski topology, via the Nullstellensatz - the radical and the topological closure both being idempotent). </p> <p>So, one answer per post, but if you have two concepts which are related, I guess it'd be okay to put them together. For the sake of the completeness (ha ha) of this list, I'll add "radical" and "topological closure".</p> <p>EDIT: My bad - I should have looked around more first. There's <a href="http://en.wikipedia.org/wiki/Closure%5F%28mathematics%29" rel="nofollow">this list</a> at Wikipedia and <a href="http://ncatlab.org/nlab/show/completion#list%5Fof%5Fcompletions%5F2" rel="nofollow">this list</a> at nLab. Well, I'm sure there's plenty more concepts out there, so if you think of any more, feel free to add them. But let's focus on how some of these concepts are related - e.g., does one kind of completion arise in terms of another? What are some general ways in which completions and closures arise?</p> http://mathoverflow.net/questions/14509/various-concepts-of-closure-or-completion-in-mathematics/14510#14510 Answer by Zev Chonoles for Various concepts of "closure" or "completion" in mathematics Zev Chonoles 2010-02-07T16:56:58Z 2010-02-07T16:56:58Z <p>The radical of an ideal - we have $\sqrt{\sqrt{I}}=\sqrt{I}$.</p> http://mathoverflow.net/questions/14509/various-concepts-of-closure-or-completion-in-mathematics/14511#14511 Answer by Zev Chonoles for Various concepts of "closure" or "completion" in mathematics Zev Chonoles 2010-02-07T16:58:43Z 2010-02-07T16:58:43Z <p>The closure of a set in a topology - we have $\overline{\bar{X}}=\bar{X}$.</p> http://mathoverflow.net/questions/14509/various-concepts-of-closure-or-completion-in-mathematics/14513#14513 Answer by Harald Hanche-Olsen for Various concepts of "closure" or "completion" in mathematics Harald Hanche-Olsen 2010-02-07T17:03:35Z 2010-02-07T17:03:35Z <p>Just about every form of compactification. The compactification of a compact space is itself, and a compactification had better be compact or it shouldn't be called a compactification.</p> <p>The same thing goes for completion of metric spaces, of course. (I know, shouldn't post two answers in one for this kind of list, but they're closely related and trivially known for everybody. I put them here for completeness' sake (no pun intended).)</p> http://mathoverflow.net/questions/14509/various-concepts-of-closure-or-completion-in-mathematics/14517#14517 Answer by Zev Chonoles for Various concepts of "closure" or "completion" in mathematics Zev Chonoles 2010-02-07T17:20:14Z 2010-02-07T17:20:14Z <p>The total ring of fractions for a ring - we have $Q(Q(R))\simeq Q(R)$.</p> http://mathoverflow.net/questions/14509/various-concepts-of-closure-or-completion-in-mathematics/14520#14520 Answer by Dinakar Muthiah for Various concepts of "closure" or "completion" in mathematics Dinakar Muthiah 2010-02-07T17:40:33Z 2010-02-07T17:40:33Z <p>Projection operators on a linear space are precisely the idempotents. All these other examples are somewhat like linear projections in that they are projecting from a category to a subcategory.</p> http://mathoverflow.net/questions/14509/various-concepts-of-closure-or-completion-in-mathematics/14533#14533 Answer by unknown (google) for Various concepts of "closure" or "completion" in mathematics unknown (google) 2010-02-07T19:01:45Z 2010-02-25T04:12:43Z <p>In general, if $A\subset B$ is a full reflective subcategory, then each object $a\in A$ is isomorphic to its image under the reflector. This seems to include many cases: $\mathbf{Ab}\subset \mathbf{Grp}$, $\mathbf{CompMet}\subset\mathbf{Met}$, <code>$\mathbf{Top}_{n+1}\subseteq \mathbf{Top}_{n}$</code> (as in <a href="http://mathoverflow.net/questions/9504/why-is-top4-a-reflective-subcategory-of-top3" rel="nofollow">http://mathoverflow.net/questions/9504/why-is-top4-a-reflective-subcategory-of-top3</a>), etc.</p> <p>EDIT: Following Pete L. Clark's comment, here is a clarification: The subcategory $A$ above is called reflective if the inclusion functor $A\subset B$ has a left adjoint, and full if this inclusion functor is full. In case $A$ is a reflective subcategory, the left adjoint to the inclusion functor is called a reflector.</p> http://mathoverflow.net/questions/14509/various-concepts-of-closure-or-completion-in-mathematics/14561#14561 Answer by Feb7 for Various concepts of "closure" or "completion" in mathematics Feb7 2010-02-07T21:54:33Z 2010-02-07T21:54:33Z <p>Ah! You edited your question! I had to delete my answer. Anyway, here is a general scenario where idempotent operations such as the one you want arise:</p> <p>In this para I am going to be vague. But the examples below given should illustrate what I have in mind.Ok, so, You have "some structure" somewhere. You want to go to the "maximal" of such a thing. You have a natural ordering on such structures you want. And also it so happens that the union of a chain of such stuff is again such a thing. Then you apply Zorn's lemma to find the maximal thing. And this operation of going and finding the maximal thing is an "idempotent completion" in your sense.</p> <p>There are plenty of examples. A few:</p> <p>$1$. A set of linearly independent vectors in a vector space is enlarged to a basis.</p> <p>$2$. An algebraic extension of a field is enlarged to the algebraic closure.</p> <p>$3$. A separable extension of a field, is enlarged to separable closure.</p> <p>$4$. A differentiable atlas on a smooth manifold is enlarged to a maximal one, ie., a differentiable structure.</p> <p>$5$. A certain functional on a Banach space is enlarged to fill the whole space, as in the proof of Hahn-Banach theorem.</p> <p>And so on, nearly in fact every application of Zorn's lemma.</p> <p>This is not a functorial way to go; but the construction as an operation is idempotent. And in some way, such as in the construction of the algebraic closure, we have an isomorphism of two such different constructions.</p> http://mathoverflow.net/questions/14509/various-concepts-of-closure-or-completion-in-mathematics/14567#14567 Answer by Haim for Various concepts of "closure" or "completion" in mathematics Haim 2010-02-07T22:58:16Z 2010-02-07T22:58:16Z <p>In Model theory we have Skolem hulls: Assuming that each formula has a corresponding Skolem function, one can take a given subset A of a model M, and close it under Skolem functions. This closure Sk(A) is the smallest elementary submodel of M containing A.</p> http://mathoverflow.net/questions/14509/various-concepts-of-closure-or-completion-in-mathematics/14572#14572 Answer by lhf for Various concepts of "closure" or "completion" in mathematics lhf 2010-02-07T23:27:34Z 2010-02-07T23:27:34Z <p>Look at <a href="http://en.wikipedia.org/wiki/Kuratowski%5Fclosure%5Faxioms" rel="nofollow">Kuratowski closures</a> for instance or <a href="http://en.wikipedia.org/wiki/Closure%5Foperator" rel="nofollow">abstract closures</a>.</p> http://mathoverflow.net/questions/14509/various-concepts-of-closure-or-completion-in-mathematics/16370#16370 Answer by Henno Brandsma for Various concepts of "closure" or "completion" in mathematics Henno Brandsma 2010-02-25T06:05:49Z 2010-02-25T06:05:49Z <p>A convex hull is also a form of closure.</p> http://mathoverflow.net/questions/14509/various-concepts-of-closure-or-completion-in-mathematics/16377#16377 Answer by gowers for Various concepts of "closure" or "completion" in mathematics gowers 2010-02-25T08:23:05Z 2010-02-25T08:23:05Z <p>The coolest closure operation I know occurs in Razborov's lower bound for the monotone circuit complexity of the clique function. In that proof he needs a class of "simple" set systems, and to get it he defines a very ingenious k-ary operation on sets and defines a simple set system to be one that is closed under that operation. I don't know what general moral to draw from that, but it's fairly different from the examples mentioned so far.</p>