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First Countable Continua and Proper Forcing

Published online by Cambridge University Press:  20 November 2018

Joan E. Hart  and
Kenneth Kunen
Joan E. Hart
Affiliation:
University ofWisconsin, Oshkosh, WI 54901, USA, hartj@uwosh.edu
Kenneth Kunen
Affiliation:
University ofWisconsin, Madison, WI 53706, USA, kunen@math.wisc.edu
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Abstract

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Assuming the Continuum Hypothesis, there is a compact, first countable, connected space of weight ${{\aleph }_{1}}$ with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add reals.

Keywords

54D0503E35connected spaceContinuum Hypothesisproper forcingirreducible map
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 3 , 01 June 2009 , pp. 604 - 616
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Bandlow, I., On the origin of new compact spaces in forcing models. Math. Nachr. 139(1988), 185–191.Google Scholar
[2] Devlin, K. J. and Shelah, S., A weak version of ♣ which follows from 2ℵ0 < 2ℵ1 . Israel J. Math. 29(1978), no. 2, 239–247.Google Scholar
[3] Dow, A., Set theory in topology. In: Recent progress in general topology, North-Holland, Amsterdam, 1992, pp. 167–197. North-Holland, 1992.Google Scholar
[4] Dow, A. and Fremlin, D., Compact sets without converging sequences in the random real model. Acta Math. Univ. Comenian. 76(2007), no. 2, 161–171.Google Scholar
[5] Džamonja, M. and Kunen, K., Measures on compact HS spaces. Fund. Math. 143(1993), no. 1, 41–54.Google Scholar
[6] Džamonja, M. and Kunen, K., Properties of the class of measure separable compact spaces. Fund. Math. 147(1995), no. 3, 261–277.Google Scholar
[7] Eisworth, T., Totally proper forcing and the Moore- Mrowka problem, Fund. Math. 177 (2003), no. 2, 121–136.Google Scholar
[8] Eisworth, T. and Roitman, J., CH with no Ostaszewski spaces. Trans. Amer. Math. Soc. 351(1999), no. 7, 2675–2693.Google Scholar
[9] Fedorchuk, V. V., A compact Hausdorff space whose infinite closed subsets are n-dimensional. Math. USSR Sb. 25(1975), no. 1, 37–57.Google Scholar
[10] Fedorchuk, V. V., Ivanov, A. V., and J. van Mill, Intermediate dimensions of products. Topology Appl. 153(2006), no. 17, 3265–3276.Google Scholar
[11] Gregory, J., A countably distributive complete Boolean algebra not uncountably representable. Proc. Amer. Math. Soc. 42(1974), 42–46.Google Scholar
[12] Hart, J. and Kunen, K., Inverse limits and function algebras. Topology Proc. 30(2006), no. 2, 501–521.Google Scholar
[13] Hellsten, A., Hyttinen, T., and Shelah, S., Potential isomorphism and semi-proper trees. Fund. Math. 175(2002), no. 2, 127–142.Google Scholar
[14] Shelah, S., Whitehead groups may be not free, even assuming CH. I. Israel J. Math. 28(1977), no. 3, 193–204.Google Scholar
[15] Sierpiński, W., Hypothèse du continu. Second edition, Chelsea Publishing Company, New York, 1956.Google Scholar