Talk:Moore space (topology)
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This article is lacking a definition. What exactly is a Moore space? -- Fropuff 03:40, 21 November 2006 (UTC)
Addressing some points
1) As suggested, there is a conspicuously lacking definition. The correct definition should be: a Moore space is a regular Hausdorff developable space. This definition first appeared in R.L. Moore's 1932 Foundations of point set theory. A MathSciNet reference for the revised edition is
- MR0150722 (27 #709) Moore, R. L. Foundations of point set theory. Revised edition. American Mathematical Society Colloquium Publications, Vol. XIII American Mathematical Society, Providence, R.I. 1962 xi+419 pp. (Reviewer: F. Burton Jones)
Further historical information can be found in the following articles:
- MR0199840 (33 #7980) Jones, F. Burton Metrization. Amer. Math. Monthly 73 1966 571--576. (Reviewer: R. W. Bagley)
- MR0203661 (34 #3510)
- Bing, R. H.
- Challenging conjectures.
- Amer. Math. Monthly 74 1967 no. 1, part II, 56--64.
Presumably Vickery's theorem has something to do with the following paper (I have included the review):
- MR0001909 (1,317f)
- Vickery, C. W.
- Axioms for Moore spaces and metric spaces.
- Bull. Amer. Math. Soc. 46, (1940). 560--564.
- The author gives a set of five axioms in terms of point and region and proves that they constitute necessary and sufficient conditions for a space to be metric and complete. A space is metric if and only if it satisfies the first three and the fifth of the axioms. The first three alone are shown to be equivalent to R. L. Moore's Axiom $1_0$ and the first four alone to Moore's Axiom 1.
2) Agreed that the non-example of the Sorgenfrey line is badly placed. In fact, that it is not a Moore space can be deduced from the results in the following section (together with the consistency of the continuum hypothesis).
3) References are needed for all of the assertions of the History section (which is not well named, being rather a survey of relatively recent results in the area).
4) I cannot see why the hedgehog space is being singled out either. Better is to emphasize the fact that all metrizable spaces are Moore spaces and that the original motivation of Moore was to present axioms for a class of spaces which contains the metrizable spaces. (Taking developability as one of your fundamental axioms for a space seems to me to be possibly brilliant and certainly bizarre...)
I don't think it really matters for Wikipedia purposes but instead of writing that a Moore space is a regular Hausdorff developable space, an equivalent definition is to say that a Moore space is a regular T_1 developable space. Or even better, one could write that a Moore space is a T_3 space that is also developable. Anyway since I am not sure about Wikipedia conventions, perhaps writing 'regular Hausdorff' will be easier to understand than any definitions that I have provided.
Any qualified reader will understand both "regular Hausdorff" and "regular T_1" and recognize their equivalence. I prefer using descriptive words rather than jargony abbreviations when possible, but "T_1" is such a well-known jargony abbreviation that it might as well be a word. The problem with "T_3" is that there is absolutely no agreement (in the worldwide mathematical community) as to whether this includes the T_1 axiom or not. The wikipedia convention is that it does include T_1, but there is no way for the reader to know this without clicking on the link.
Topo, you cut and paste the references I provided without properly formatting them or even pruning off the Math Reviews information. (Also there is no longer any reference to Vickery's theorem in the article, so either this should be put back or -- better, in my opinion -- that reference should not be included.) Correctly formatting references may not be much fun, but it is part of the job. Plclark (talk) 05:03, 29 July 2008 (UTC)Plclark
According to another source (planet math), Moore spaces are regular developable spaces having nested developments. Is this a stronger requirement compared to that given by the article?
Ooops! Vickery's theorem implies that both definitions are equivalent.