Talk:Completely Hausdorff space
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I have written this in line with the PlanetMath source; but see completely Hausdorff space. The terminology seems to be somewhat confused, and perhaps merges should go on.
Charles Matthews 09:26, 29 Dec 2004 (UTC)
We now have an internal inconsistency that needs addressing. This page and completely Hausdorff space currently describe the same concept:
- a space in which points can be separated by a function
and we have no page describing
- a space in which points can be separated by closed neighborhoods.
Different authors seem to have different views on which name goes with which. Currently, our separation axioms page says completely Hausdorff = separated by a function and Urysohn = separated by closed nbds. The PlanetMath article reserves these. I don't know which is the more common or more modern definition but I submit that we should at least be internally consistent within Wikipedia. Before I run around changing everything I think we should get on consensus on which naming convention to use. I'll start a table outlining which authors/sources use which naming convention. Please add your input if you have any. -- Fropuff 05:25, 2005 Feb 3 (UTC)
| completely Hausdorff |
T2½ | functionally Hausdorff |
Urysohn | |
|---|---|---|---|---|
| Separation axiom | func | c.nbd | c.nbd | |
| PlanetMath | c.nbd | c.nbd | func | func |
| Willard, General Topology (1970) |
func | func | c.nbd | |
| Steen and Seebach (1970) | c.nbd | c.nbd | func | |
| HarperCollins Dictionary of Mathematics (1991) |
c.nbd | c.nbd |
*In this table func means "points separated by a function" and c.nbd means "points separated by closed neighborhoods".
My first suggestion is that both these concepts — whatever we decide to call them — should be discussed on the same page. -- Fropuff 05:35, 2005 Feb 3 (UTC)
I think that your [Fropuff;s] "first suggestion" above is a pretty good place to start; that gets everything together on one page, so that when people spread out from there, then they're not confused. I have some more references to add to this table, at least some of which I should be able to do this evening (California time). AFAICR, my only reason for picking the definitions that I did on Separation axiom is that Willard is generally more modern in tone than Steen & Seebach (despite the coincidence of their dates), but further references may (or may not) confirm that Willard's terminology really is more modern. In any case, my position is that WP should favour the modern references (if that is sensible), while of course explaining the variations for purposes of NPOV. Personally, I do find Willard's terminology more logical, since it follows the patterns set by "completely regular space"; logic is also the reason that I doubt that there will be any variation in the meaning of "T2½" (see the chart that you improved in Separation axiom#Relationships between the axioms). -- Toby Bartels 00:55, 2005 Feb 8 (UTC)
-
- Sorry, I modified Urysohn space without looking at the talk here. I thought it was just a matter of wording (and the wording was bad). So I fixed the wording, made the mistake of comparing things with PlanetMath seeing that all is fine, and removed the attention tag. Now I will stay tuned with what you decide. My knowledge of topology is such that I cannot well separate in my mind T2½ from T2 (pun intended). Oleg Alexandrov 15:47, 10 Feb 2005 (UTC)
No worries, the attention tag was a little ambiguous, and your mistake completely understandable. I've now added what I hope is a more descriptive notice. I will leave this discussion open for a few weeks to see if anyone else has any input, before I try and patch things up. -- Fropuff 16:44, 2005 Feb 10 (UTC)
- For what its worth I vote for Willard's usage, but with caveats.
- I've polled a couple of eminent topologists (my PHD advisor and a distinguished German collegue of his) and they both agree that Willard's definitions are more standard. And a little Googleing seems to confirm this. See for example: [1].
- But I also find many writers who use the Steen and Seebach definitions, not surprising when you consider how influential Counterexamples has been. And there is certain logic behind that usage since a Urysohn function is defined (by both Willard and S&S, and AFAIK, every one else) to be the "separating function" provided by Urysohn's lemma: A space X is normal iff whenever A and B are disjoint closed sets in X, there is a continuous function f:X-->[0,1] with f(A) = 0 and f(B) = 1.
- Also to make matters a bit more complicated, note: there is also the different notion of "the Urysohn space" (also "the universal Urysohn space"), for the unique complete separable metric space that is universal and homogeneous, (defined, apparently, by Urysohn in his last paper in 1924). And there also seems to be something some writers are calling a "continuously Urysohn space".
- If anybody has access to Kelley's General Topology, I'd like to know what he says.
- Paul August ☎ 19:58, Feb 10, 2005 (UTC)
I checked Kelley, but he stops short of defining any of these, defining only Hausdorff, regular, and completely regular. I agree that both naming conventions are logical (by analogy with either Urysohn's lemma or completely regular space). For the record, I favor Willard's definition as well. -- Fropuff 20:53, 2005 Feb 10 (UTC)
Okay, since no one else seems to have any input, I'll revert everything to Willard's definition and put everything on the completely Hausdorff space page. Thanks for your input Toby and Paul. -- Fropuff 02:36, 2005 Mar 4 (UTC)
