Extending real-valued functions in βκ

Alan Dow

Fundamenta Mathematicae (1997)

  • Volume: 152, Issue: 1, page 21-41
  • ISSN: 0016-2736

Abstract

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An Open Coloring Axiom type principle is formulated for uncountable cardinals and is shown to be a consequence of the Proper Forcing Axiom. Several applications are found. We also study dense C*-embedded subspaces of ω*, showing that there can be such sets of cardinality c and that it is consistent that ω*{pis C*-embedded for some but not all p ∈ ω*.

How to cite

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Dow, Alan. "Extending real-valued functions in βκ." Fundamenta Mathematicae 152.1 (1997): 21-41. <http://eudml.org/doc/212197>.

@article{Dow1997,
abstract = {An Open Coloring Axiom type principle is formulated for uncountable cardinals and is shown to be a consequence of the Proper Forcing Axiom. Several applications are found. We also study dense C*-embedded subspaces of ω*, showing that there can be such sets of cardinality $c$ and that it is consistent that ω*\{pis C*-embedded for some but not all p ∈ ω*. },
author = {Dow, Alan},
journal = {Fundamenta Mathematicae},
keywords = {Proper Forcing Axiom; Open Colouring Axiom; -embedding; Čech-Stone compactification; first-countability; countable separability; reflection theorem for ideals of countable sets; Hausdorff-Luzin-type gap; finite-to-one graph type family; countably tight space; Martin's Axiom},
language = {eng},
number = {1},
pages = {21-41},
title = {Extending real-valued functions in βκ},
url = {http://eudml.org/doc/212197},
volume = {152},
year = {1997},
}

TY - JOUR
AU - Dow, Alan
TI - Extending real-valued functions in βκ
JO - Fundamenta Mathematicae
PY - 1997
VL - 152
IS - 1
SP - 21
EP - 41
AB - An Open Coloring Axiom type principle is formulated for uncountable cardinals and is shown to be a consequence of the Proper Forcing Axiom. Several applications are found. We also study dense C*-embedded subspaces of ω*, showing that there can be such sets of cardinality $c$ and that it is consistent that ω*{pis C*-embedded for some but not all p ∈ ω*.
LA - eng
KW - Proper Forcing Axiom; Open Colouring Axiom; -embedding; Čech-Stone compactification; first-countability; countable separability; reflection theorem for ideals of countable sets; Hausdorff-Luzin-type gap; finite-to-one graph type family; countably tight space; Martin's Axiom
UR - http://eudml.org/doc/212197
ER -

References

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  2. [BS87] A. Blass and S. Shelah, There may be simple P ℵ 1 and P ℵ 2 -points and the Rudin-Keisler order may be downward directed, Ann. Pure Appl. Logic 83 (1987), 213-243. Zbl0634.03047
  3. [vDKvM] E. K. van Douwen, K. Kunen and J. van Mill, There can be proper dense C*-embedded subspaces in βω-ω, Proc. Amer. Math. Soc. 105 (1989), 462-470. Zbl0685.54001
  4. [Dow88] A. Dow, An introduction to applications of elementary submodels to topology, Topology Proc. 13 (1988), 17-72. Zbl0696.03024
  5. [Dow92] A. Dow, Set theory in topology, in: Recent Progress in General Topology, M. Hušek and J. van Mill (eds.), Elsevier, 1992, 169-197. 
  6. [DJW89] A. Dow, I. Juhász and W. Weiss, Integer-valued functions and increasing unions of first countable spaces, Israel J. Math. 67 (1989), 181-192. Zbl0711.54002
  7. [DM90] A. Dow and J. Merrill, ω 2 * - U ( ω 2 ) can be C*-embedded in β ω 2 , Topology Appl. 35 (1990), 163-175. 
  8. [HvM90] K. P. Hart and J. van Mill, Open problems on βω, in: Open Problems in Topology, J. van Mill and G. M. Reed (eds.), North-Holland, 1990, 97-125. 
  9. [Laf93] C. Laflamme, Bounding and dominating number of families of functions on ω, Math. Logic Quart. 40 (1994), 207-223. Zbl0835.03015
  10. [Mil84] A. Miller, Rational perfect set forcing, in: Axiomatic Set Theory, Contemp. Math. 31, 1984, 143-159. 
  11. [She84] S. Shelah, On cardinal invariants of the continuum, Axiomatic Set Theory, Contemp. Math. 31, 1984, 183-207. 
  12. [Tod89a] S. Todorčević, Partition Problems in Topology, Contemp. Math. 84, Amer. Math. Soc., 1989. 
  13. [Tod89b] S. Todorčević, Tightness in products, Interim Rep. Prague Topolog. Sympos. 4 (1989), 7-8. 

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