If it looks and smells like the reals...

Franklin Tall

Fundamenta Mathematicae (2000)

  • Volume: 163, Issue: 1, page 1-11
  • ISSN: 0016-2736

Abstract

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Given a topological space ⟨X,T⟩ ∈ M, an elementary submodel of set theory, we define X M to be X ∩ M with topology generated by U ∩ M:U ∈ T ∩ M. We prove that if X M is homeomorphic to ℝ, then X = X M . The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted.

How to cite

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Tall, Franklin. "If it looks and smells like the reals...." Fundamenta Mathematicae 163.1 (2000): 1-11. <http://eudml.org/doc/212426>.

@article{Tall2000,
abstract = {Given a topological space ⟨X,T⟩ ∈ M, an elementary submodel of set theory, we define $X_M$ to be X ∩ M with topology generated by U ∩ M:U ∈ T ∩ M. We prove that if $X_M$ is homeomorphic to ℝ, then $X = X_M$. The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted.},
author = {Tall, Franklin},
journal = {Fundamenta Mathematicae},
keywords = {elementary submodel; real line; locally compact separable metric space; local compactness; elementary submodel of set theory; separable metric spaces},
language = {eng},
number = {1},
pages = {1-11},
title = {If it looks and smells like the reals...},
url = {http://eudml.org/doc/212426},
volume = {163},
year = {2000},
}

TY - JOUR
AU - Tall, Franklin
TI - If it looks and smells like the reals...
JO - Fundamenta Mathematicae
PY - 2000
VL - 163
IS - 1
SP - 1
EP - 11
AB - Given a topological space ⟨X,T⟩ ∈ M, an elementary submodel of set theory, we define $X_M$ to be X ∩ M with topology generated by U ∩ M:U ∈ T ∩ M. We prove that if $X_M$ is homeomorphic to ℝ, then $X = X_M$. The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted.
LA - eng
KW - elementary submodel; real line; locally compact separable metric space; local compactness; elementary submodel of set theory; separable metric spaces
UR - http://eudml.org/doc/212426
ER -

References

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  12. [KT] K. Kunen and F. D. Tall, The real line in elementary submodels of set theory, J. Symbolic Logic, to appear. Zbl0960.03033
  13. [R] J. Roitman, Basic S and L, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, Amsterdam, 1984, 295-326. 
  14. [S] J. H. Silver, The independence of Kurepa's conjecture and two-cardinal conjectures in model theory, in: Axiomatic Set Theory, D. S. Scott (ed.), Proc. Sympos. Pure Math. 13, Amer. Math. Soc., Providence, 1971, 383-390. Zbl0255.02068
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