Talk:Locally compact space

message: Well, I've gone and changed the definition of local compactness again. Actually, the article now mentions all three definitions up front and refuses to commit to any one. I did this because almost all references I've checked were not using our (previous) definition. The sole exception being Willard. I see no good reason to favor Willard's definition above the rest (I welcome arguments either way). If Wikipedia is to be accurate it should reflect the definitions that readers are likely to encounter in the "wild". Unfortunately, this seems to preclude choosing a favored definition here. -- Fropuff 02:49, 17 October 2006 (UTC)
thread_title: Definition

message: this page never describes, even vaguely, what a compact space is. an article should not require another article for a modicum of understanding. 66.160.66.86 (talk) 16:15, 22 January 2008 (UTC)
thread_title:  expand 

message: The article says: The one-point compactification of the rational numbers Q is compact and therefore locally compact in senses (1) and (2) but it is not locally compact in sense (3).
But one-point compactification is defined only for locally compact spaces, where Q is (as mentioned some not locally compact. So, what does this sentence wants to say? -- Paul Ebermann (talk) 14:36, 8 July 2008 (UTC)
replies: text: The construction should not require the original space to be locally compact. If Y is a topological space, define the Alexandroff extension Y* to be Y union {p}, with neighborhoods of p defined as (Y-L) union {p} where L is compact.  Y* is compact, Y is contained as a subspace, Y is open in Y*, Y is dense in Y* if and only if Y is not compact, Y* is Hausdorff if and only if Y is locally compact and Hausdorff.  This is from Willard, 1970, 19A, p. 140. JackSchmidt (talk) 15:45, 8 July 2008 (UTC)
replies: text: The construction does not, but the one-point compactification may no longer be Hausdorff.  Indeed, the one-point compactification of the rationals is not Hausdorff, since any neighborhood of the point at infinity is a finite-complement neighborhood.  This should probably be addressed somewhere in the one-point compactification article, but I am reluctant to omit the requirement of local compactness from the definition.  Perhaps it should be mentioned in a separate "generalizations" section to avoid polluting the main results with pathologies.  siℓℓy rabbit (talk) 16:00, 8 July 2008 (UTC)
thread_title:  "one-point compactification of the rational numbers" 

message: I am having some issues with some of the examples of "Hausdorff Spaces that are not Locally Compact" on this page, see for instance:
: * the space Q of rational numbers (endowed with the topology from R), since its compact subsets all have empty interior and therefore are not neighborhoods;
As far as i'm aware, if some subset of \mathbb{Q} is taken as a subset of \mathbb{R} then sure, its interior is empty. But *within* the sub-space topology the interior of, e.g. [a,b] is just (a,b), no? A valid reason that \mathbb{Q} is not locally compact is maybe because sequences of the form \left( \left[ r + \frac{\lfloor 2^n \sqrt{2} \rfloor}{2^n},r + \frac{\lfloor 2^n \sqrt{2} \rfloor + 1}{2^n} \right] \right)_{n \in \mathbb{N}, n > N} for suitable rational r and number N form descending sequences of closed non-empty sets with empty interesection in \mathbb{Q}, and by suitable choice of N and r can be placed inside any neighborhood, showing that no neighborhood is compact.
... and on this example:
: * the subspace {(0,0)} union {(x,y) : x > 0} of R2, since the origin does not have a compact neighborhood;
I'm wondering how this can be? Is there a citation (specific chapter in one of the standard books)? It seems that this is a closed subset of R and hence locally compact? Am i missing something?  
199.43.12.100 (talk) 10:45, 25 April 2013 (UTC)
replies: text: In the first example, you are right that if a and b are rationals with a < b, then the interior (in Q) of [a,b] ∩ Q is (a,b) ∩ Q. But [a,b] ∩ Q is not compact, so this doesn't contradict the assertion. As for the other example, it's not a closed subset of R2. --Zundark (talk) 11:46, 25 April 2013 (UTC)
thread_title:  Examples Correct?