How separable is a space? That depends on your set theory!

Author:
Franklin D. Tall

Journal:
Proc. Amer. Math. Soc. **46** (1974), 310-314

MSC:
Primary 54B10; Secondary 54E30

MathSciNet review:
0362188

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Abstract | References | Similar Articles | Additional Information

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 set-theoretic consistency and independence of the proposition that this property is preserved by products of factors is established.

**[1]**R. H. Bing,*Metrization of topological spaces*, Canadian J. Math.**3**(1951), 175–186. MR**0043449****[2]**D. D. Booth,*Countably indexed ultrafilters*, Thesis, University of Wisconsin, Madison, Wis., 1969.**[3]**L. Bukovský,*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, 1967, pp. 83–86. MR**0231723****[4]**W. W. Comfort,*A short proof of Marczewski’s separability theorem*, Amer. Math. Monthly**76**(1969), 1041–1042. MR**0248742****[5]**N. E. Foland and R. B. Kirk,*Products of spaces with -dense subsets*(preprint).**[6]**K. Gödel,*Consistency-proof for the generalized continuum hypothesis*, Proc. Nat. Acad. Sci. U.S.A.**25**(1939), 220-224.**[7]**R. W. Heath,*Screenability, pointwise paracompactness, and metrization of Moore spaces*, Canad. J. Math.**16**(1964), 763–770. MR**0166760****[8]**I. Juhász,*Cardinal functions in topology*, Mathematical Centre, Amsterdam, 1971.**[9]**D. A. Martin and R. M. Solovay,*Internal Cohen extensions*, Ann. Math. Logic**2**(1970), no. 2, 143–178. MR**0270904****[10]**Fritz Rothberger,*On some problems of Hausdorff and of Sierpiński*, Fund. Math.**35**(1948), 29–46. MR**0029958****[11]**R. M. Solovay and S. Tennenbaum,*Iterated Cohen extensions and Souslin’s problem*, Ann. of Math. (2)**94**(1971), 201–245. MR**0294139****[12]**F. D. Tall,*An alternative to the continuum hypothesis and its uses in general topology*(preprint).**[13]**Albert Wilansky,*Research Problems: How Separable is a Space?*, Amer. Math. Monthly**79**(1972), no. 7, 764–765. MR**1536789**, 10.2307/2316270

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0362188-5

Keywords:
Separable,
sequential limit,
Martin's axiom,
topological product,
set-theoretic consistency

Article copyright:
© Copyright 1974
American Mathematical Society