How separable is a space? That depends on your set theory!
Author:
Franklin D. Tall
Journal:
Proc. Amer. Math. Soc. 46 (1974), 310314
MSC:
Primary 54B10; Secondary 54E30
MathSciNet review:
0362188
Fulltext PDF Free Access
Abstract 
References 
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Abstract: A. Wilansky has raised the question of the productive behaviour of the property of having a countable set, such that each point is a sequential limit point of the set. The settheoretic consistency and independence of the proposition that this property is preserved by products of factors is established.
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W. Heath, Screenability, pointwise paracompactness, and metrization
of Moore spaces, Canad. J. Math. 16 (1964),
763–770. MR 0166760
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I. Juhász, Cardinal functions in topology, Mathematical Centre, Amsterdam, 1971.
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1536789, http://dx.doi.org/10.2307/2316270
 [1]
 R. H. Bing, Metrization of topological spaces, Canad. J. Math. 3 (1951), 175186. MR 13, 264. MR 0043449 (13:264f)
 [2]
 D. D. Booth, Countably indexed ultrafilters, Thesis, University of Wisconsin, Madison, Wis., 1969.
 [3]
 L. Bukovsky, Borel subsets of metric separable spaces, General Topology and its Relations to Modern Analysis and Algebra II, Proc. Second Prague Topological Sympos., 1966, Academia, Prague; Academic Press, New York, 1967, pp. 8386. MR 38 #51. MR 0231723 (38:51)
 [4]
 W. W. Comfort, A short proof of Marczewski's separability theorem, Amer. Math. Monthly 76 (1969), 10411042. MR 40 #1993. MR 0248742 (40:1993)
 [5]
 N. E. Foland and R. B. Kirk, Products of spaces with dense subsets (preprint).
 [6]
 K. Gödel, Consistencyproof for the generalized continuum hypothesis, Proc. Nat. Acad. Sci. U.S.A. 25 (1939), 220224.
 [7]
 R. W. Heath, Screenability, pointwise paracompactness and metrization of Moore spaces, Canad. J. Math. 16 (1964), 763770. MR 29 #4033. MR 0166760 (29:4033)
 [8]
 I. Juhász, Cardinal functions in topology, Mathematical Centre, Amsterdam, 1971.
 [9]
 D. Martin and R. M. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), 143178. MR 42 #5787. MR 0270904 (42:5787)
 [10]
 F. Rothberger, On some problems of Hausdorff and of Sierpiński, Fund. Math. 35 (1948), 2946. MR 10, 689. MR 0029958 (10:689a)
 [11]
 R. M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin's problem, Ann. of Math. (2) 94 (1971), 201245. MR 45 #3212. MR 0294139 (45:3212)
 [12]
 F. D. Tall, An alternative to the continuum hypothesis and its uses in general topology (preprint).
 [13]
 A. Wilansky, How separable is a space? Amer. Math. Monthly 79 (1972), 764765. MR 1536789
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197403621885
PII:
S 00029939(1974)03621885
Keywords:
Separable,
sequential limit,
Martin's axiom,
topological product,
settheoretic consistency
Article copyright:
© Copyright 1974
American Mathematical Society
