Talk:End (topology)

Should mention that under suitable conditions, the union of a space with its ends is compact. See Spivak's /Differential Geometry/ vol. 1 exercises to the first chapter for details. 128.135.100.161 23:42, 10 March 2007 (UTC)

my favorite characterization
I think the nicest definition of an end of a space X is a function e that assigns to every compact subset K of X a connected component e(K) of $$X\setminus K$$ such that $$e(K')\subset e(K)$$ whenever $$K'\supset K$$. Anyone knows if and where this appears in the literature? Oded (talk) 06:46, 3 August 2008 (UTC)

Dear Oded,

I don't think that this definition appears in literature because your definition is not really an end. For instance, take an uncountable space endowed with the countable complement topology (or sometimes referred to as the co-countable topology). Note that every open subspace of this space is uncountable and hence connected. If e is the function that maps each compact set onto its complement then e satisfies your condition (A subspace of an uncountable set with the co-countable topology is compact iff it is finite. Therefore, if e maps a compact set K onto its complement, its complement will be uncountable and hence connected. Therefore, e maps each compact set onto a connected component of its complement. Also, it is clear that $$e(K')\subset e(K)$$ whenever $$K'\supset K$$). But e is not an end since the closure of each neighbourhood of an end is the whole space (since this space is hyperconnected and every open set is thus dense) so that the intersection of the closures of such neighbourhoods will be the whole space. This intersection is therefore not empty.

Q.E.D

Topology Expert (talk) 10:59, 11 August 2008 (UTC)


 * This means that there is another inconsistency in the article. I figured that the definition I gave above is just a different (more elementary) way to state the definition given as an inverse limit (which appears further down in the article). Therefore, there are some assumptions necessary to make these definitions equivalent. We have to sort out the definitions given in the literature and rewrite the article. Oded (talk) 15:30, 11 August 2008 (UTC)

I wrote this on the talk page of User: OdedSchramm but I feel that this should appear here:

I have found requirements to make the two definitions equivalent; they are equivalent for Hausdorff, σ-locally compact spaces. Whenever I say 'your definition' I just mean the definition that you re-formulated from the article.

Proof:

Let X satisfy the given hypothesis. For each point, x in X, choose a compact subset Cx of X that contains a neighbourhood Ux of x. The collection of all {Ux} [indexed by the space X] forms an open cover of X and therefore has a countable subcover {Ux i } for i in N. Choose for each element in the subcover, an element Cx i  containing it. Then X equals the union of all Cx i  for i in N. Let An equal the union of all such Cx i  for i varying from 1 to n. If e is the function that maps each An onto its complement, then we assert e satisfies the condition of an end. Since X is Hausdorff, each An is closed so that its complement is open. If Vi equals to the union of all Ux i  for i varies from 1 to n, then the union of all Vi for i in N is X. Note also that the closure of the complement of each An is a subset of the complement of Vi which means that the intersection over all n of the closure of the complement of An is empty. Hence the intersection of the closures of all sets in the range of e will be empty. Therefore, e is an end as desired.

Q.E.D

For an uncountable set given the co-countable topology, there exists no function satisfying your hypothesis that is an end since there is only one such function. Whereas, in my proof I have shown the existence of a function satisfing your hypothesis that is still an end.

The domain of the function constructed in the proof is a subset of the collection of all compact sets but it still is an end. Perhaps your definition should allow functions that are defined on any subcollection of the collection of all compact subsets of X. In fact, an end is a countable collection of sets; according to your definition an end is a function defined on all compact subsets of X and therefore need not be countable.

Therefore, I am led to believe that your definition has to be modified quite significantly in order to agree with the definition given in the article. These are the possible modifications necessary:


 * An end should be a function defined on a countable subcollection of the collection of all compact subsets of X


 * You said that an end is a function that assigns to every compact set K a connected component e(K) of $$X\setminus K$$; you are assuming that the component of the complement of K is open which need not be the case because:

a) In locally connected spaces, components of open sets are open. If $$X\setminus K$$ is open, then X has to be locally connected to ensure that e(K) is really a neighbourhood of an end.

b) Also, whoever assumed that $$X\setminus K$$ is open? It is not open unless compact subsets of X are closed, i.e the Hausdorff condition must be assumed.

After these two modifications are made, your definition of an end will agree with the article's definition for σ-locally compact spaces.

I am not very knowledgeable about ends but the article says that an end may be used to 'compactify' a space. If this is true, then adding to compactness-related axioms to your definition may not be the best thing to do for the article's purposes. Do you have any opinions on this? In my opinion the definition of an end through functions should be removed since the original definition given by the article is the simplest and the most appropriate.

Topology Expert (talk) 08:27, 12 August 2008 (UTC)


 * We should figure out which definition is more prevalent in the literature and use that one. Other definitions appearing in the literature should also be mentioned with an explanation as to the conditions that are required to make them equivalent. I think the inverse limit definition is widely used. I have not seen before the definition appearing as the main definition in the article now. (But that does not mean it is not widely used.) It is useful to have the set of ends defined as a topological space, not just as a set, and it is also useful to have the ends provide a compactification of the space. It is reasonable that this can only be achieved under certain assumptions. Have ends been useful in any non-Hausdorff context? Oded (talk) 17:41, 12 August 2008 (UTC)


 * set-theoretic and algebraic terms like almost-invariant set are used also but they are not so common. You can read at the Daniel Cohen's article Ends and free product groups this idea on line at --kmath (talk) 01:29, 29 August 2008 (UTC)

seems wrong to me
The characterization in terms of rays seems wrong. Homotopy is related to path-connectedness, but the notion of an end is related to connectedness. For example, if we have a space in which the path-connected components are all points, then there are no nontrivial homotopies. However, there are connected sets, and the space may have just one end. Oded (talk) 06:42, 3 August 2008 (UTC)
 * Yes, here there is something weird but at least for path-connected, strongly locally finite CW-complex, this is valid as is seen in "Topological methods in group theory" of Ross Geoghegan, page 295,--kmath (talk) 06:55, 3 August 2008 (UTC)
 * Do you mind putting in the version that appears there? I don't have access to a library in the next week or so. Oded (talk) 12:55, 3 August 2008 (UTC)


 * 13.4 Ends of space
 * Throughout this section Y denotes a path-connected, strongly locally finite CW complex. A proper ray in Y is a proper map $$\omega:[0,\infty)\to Y$$. Two proper rays $$\omega_1$$ and $$\omega_2$$ in Y define the same end of Y if $$\omega_1|\mathbb{N}$$ and  $$\omega_2|\mathbb{N}$$ are properly homotopic; here, $$\mathbb{N}=\{0,1,2,...\}$$ considered as a discrete subspace of $$[0,\infty)$$. This is an equivalence relation; an end of Y is an equivalence class of proper rays. The set of ends of Y is denoted by $$\mathcal{E}(Y)$$.
 * ... this is the 1st paragraph at p.295,--kmath (talk) 01:33, 4 August 2008 (UTC)


 * Thanks. I guess this does establish that this is a notion of the set of ends. However, do we actually know that this notion agrees with the other definitions in this setting? Is that proved in that book? If not, we could try to come up with a proof, but we are not supposed to per WP:OR. If this is really trivial, then perhaps it is no big deal. Alternatively, a good solution is just to say that a related notion for path-connected ... CW complexes is ..., as given by ... Oded (talk) 05:20, 4 August 2008 (UTC)
 * Look, my friend, I wouldn't worry about WP:OR cuz anyhow this subjet is technical enough to scare almost to everyone who lurks here... anyway i'm going to check your question but leisurely because we are vacationing, right? If you what an example about the why my opinion on WP:OR see the discussion on Alexander trick, see you later...--kmath (talk) 05:55, 5 August 2008 (UTC)
 * especifically look at ;--kmath (talk) 06:19, 5 August 2008 (UTC)
 * What does that last link have to do with WP:OR? FWIW, I strongly suggest we stick to referenced sources.  I have a copy of Hughes and Ranicki's "The Ends of Complexes", which I consider to be the definitive monograph on the subject.  When I get some time, I'll look in there and try to supplement this article.  In the meantime, I recommend that references be found for anything that you wish to add to the article.  VectorPosse (talk) 10:23, 5 August 2008 (UTC)
 * My point about WP:OR is showing with the example -there in Alexander trick- that for some people some topics seem "too advanced", but being standard, i.e. depends on subjetivity... am i wrong? In the other hand thanks for the hint: Ends of complexes which -i found- can be consulted at .--kmath (talk) 17:57, 5 August 2008 (UTC)
 * Why am I so uptight about OR? Well, I had some disagreements with an editor who I suspected to make a habit of proving some facts (not particularly to my personal taste) and putting them on advanced math WP articles. Who am I to judge what should be included and what should not? I figured that the WP:OR is the only reasonable criterion to apply. Oded (talk) 01:21, 6 August 2008 (UTC)
 * It is true that "too advanced" is subjective. That's why it isn't the inclusion criterion.  "Referenced", however, is not subjective.  Oded makes a very good point; requiring our articles to be sourced is the only way to make sure the cranks don't take over.  :)  VectorPosse (talk) 10:12, 6 August 2008 (UTC)

how to improve the article?
My own personal exposure to the subject of ends is mostly in the realm of ends of graphs and ends of trees. In that setting, the concept is simpler and easy to visualize. I think it would be a good idea to include a discussion of ends of trees and of graphs as illustrative examples.

Anyone has a suggestion for an example theorem which uses the concept of ends in its proof (but not in its statement)? If there is something like that which could be reasonably easy to explain, that would help motivate the subject. Sometimes the best explanation of a subject is an example showing how useful it can be. Oded (talk) 01:29, 6 August 2008 (UTC)
 * Agree, why don't you enlighten us?...
 * I'll put it on my to-do list (but it will take a while) Oded (talk) 02:58, 7 August 2008 (UTC)
 * and as i understand John Stallings gave one of the first applications into topology, especifically he gave another proof of the difficult Papakiriakopoulus' sphere theorem (3-manifolds). Details are in his Group theory and three-dimensional manifolds of 1971, which, unfortunately, i don't have yet... but it don't be long that i have it :) —Preceding unsigned comment added by Juan Marquez (talk • contribs) 02:56, 6 August 2008 (UTC)


 * Great. I looked up the John Stallings article. The result mentioned there about a characterization of the finitely generated groups that have more than one end sounds familiar. But doesn't the statement there miss some trivial assumption? Perhaps I'm confused, but isn't $$\mathbb Z$$ a counterexample to the theorem as stated there? Oded (talk) 03:20, 6 August 2008 (UTC)
 * Ok, $$\mathbb{Z}$$ is finitely generated and it has two ends, but: does split over finite groups? --kmath (talk) 03:47, 6 August 2008 (UTC)
 * Response: Yes; $$\scriptstyle \mathbb{Z}=1*_1=\langle 1,t|t^{-1}1t=1\rangle=\langle t|\ \rangle=F_1$$ corresponding to the fundamental group of the graph of groups of a point and a vertex both with a trivial group attached, wow!--kmath (talk) 01:51, 31 August 2008 (UTC)

End compactification
I don't understand why the space $$X$$ has to be connected and locally connected for the end compactification be compact. It seems to be wrong : $$]0,1] \cup [2,3[$$ (as a subspace of the real line) is locally connected but not connected ; it has two ends, and its end compactification is homeomorphic to $$[0,1] \cup [2,3]$$ which is compact. — Preceding unsigned comment added by TopoGeo (talk • contribs) 08:21, 19 October 2011 (UTC)