Definition talk:Closure (Topology)

In this case, I can't see how there is merit for two separate pages. The derived set $H'$ of $H$ is defined as the set of all limit points of $H$. Comments? --abcxyz (talk) 01:11, 16 November 2012 (UTC)


 * Someone who does not connect those two facts may not realise the two things are the same till it's pointed out in a page demonstrating their equivalence. --prime mover (talk) 06:32, 16 November 2012 (UTC)


 * Last Tuesday, a second-year student told me he had learned that $\Z_n \cong \Z / n \Z$ (through an isomorphism theorem), to which statement I looked at him in utter confusion, incapable of comprehending any more why such a statement would require a proof... It's hard to imagine not knowing such things when one in fact does. --Lord_Farin (talk) 12:04, 16 November 2012 (UTC)


 * Sorry, but I still don't get it. Anyone who follows the link to Definition:Derived Set will surely see the connection, no? --abcxyz (talk) 16:23, 16 November 2012 (UTC)


 * Now that I've actually bothered to look at the page, I agree with abcxyz that this is artificial separation of definitions. They can be merged, derived set mentioned, and we'll be done with it, on to more important/interesting stuff. --Lord_Farin (talk) 16:25, 16 November 2012 (UTC)

Rewrite thoughts: The current choice of definition 1 is unfortunate. The most important definitions, I think, are: --Dfeuer (talk) 23:29, 27 February 2013 (UTC)
 * Definition 2: the intersection of containing closed sets
 * Definition 3: the smallest containing closed set
 * Definition .... NOT HERE: the set of all adherent points of the set.


 * Another one that's missing is that the closure is the complement of the interior of the complement. --Dfeuer (talk) 23:53, 27 February 2013 (UTC)


 * The last one is really not how anyone defines the closure (that I know of, I should add). Penultimate is effectively on Definition:Adherent Point. Will set out to compose a master plan. &mdash; Lord_Farin (talk) 09:21, 28 February 2013 (UTC)

Proposed scheme (after browsing through a selection of topology works):


 * Def 1: Intersection of containing closed sets
 * Def 2: Smallest containing closed set
 * Def 3: Union with limit points
 * Def 4: Set of adherent points

The following "definitions" could best be relegated to a separate page (st. like "Characterization of Topological Closure", have to think on that) because no sensible soul would define the closure in this way (e.g. the latter makes $H \subseteq H^-$ less immediate, boundary presently not defined without mentioning closure (and I haven't seen a definition for boundary that pulls that trick off)):


 * Def 5: Union with boundary
 * Def 6: Union of isolated and limit points

One could say that the same would hold for Def. 4; I found a source giving Def. 4 (J. Dugundji, Topology, 1966) so that's out of the question. &mdash; Lord_Farin (talk) 09:56, 28 February 2013 (UTC)


 * Def 4 is easier to handle than def3, at least mentally. would you mind swapping them terribly much? --Dfeuer (talk) 10:12, 28 February 2013 (UTC)


 * Yes, for Def 3 is presently the only one with sources (three). It thence does not deserve to be at the bottom. &mdash; Lord_Farin (talk) 10:46, 28 February 2013 (UTC)


 * All right. You win that one I guess. Munkres defines it as the intersection of closed containing sets (chapter 2: section 17). Kelley does the same (Chapter 1: Closure) --Dfeuer (talk) 15:03, 28 February 2013 (UTC)

You're probably wondering what I'm waiting for. The answer is as mundane as not having enough time to do it properly before the real world will suck me up again :). Hopefully I'll get round to it later tonight. &mdash; Lord_Farin (talk) 18:02, 28 February 2013 (UTC)

Closure as Set of Adherent Points
In both Mendelson 3rd edition and Topology by W.W. Fairchild & C. Ionecu Tulcea, the closure is defined as the set of adherent points (having defined previously defined adherent point independently of closure).

Unfortunately, some sources, Book:Ephraim J. Borowski/Dictionary of Mathematics and Book:Christopher Clapham/The Concise Oxford Dictionary of Mathematics/Fifth Edition define adherent point as an element of the closure (having defined the closure independently of adherent point)

If I naively include the Mendelson definition as an additional definition of the closure then I create the following situation:


 * Definition A has
 * Definition 1
 * Definition 2
 * Definition n which depends on Definition B
 * Definition n which depends on Definition B


 * Definition B has
 * Definition 1
 * Definition 2
 * Definition m which depends on Definition A
 * Definition m which depends on Definition A

While I don't think this is wrong, it might be confusing. I assume that similar situations have occurred elsewhere. How has this been resolved? --Leigh.Samphier (talk) 09:05, 11 May 2020 (EDT)