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Rothberger's property and partition relations

Published online by Cambridge University Press:  12 March 2014

Marion Scheepers
Marion Scheepers*
Affiliation:
Department of Mathematics, Boise State University, Boise, Idaho 83725, USA, E-mail: marion@math.idbsu.edu
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Extract

Let X be an infinite but separable metric space. An open cover of X is said to be large if for each x ϵ X the set {U ϵ : x ϵ U} is infinite. The symbol Λ denotes the collection of large open covers of X. An open cover of X is said to be an ω-cover if for each finite subset F of X there is a U ϵ such that F ⊆ U, and X is not a member of , X is said to have Rothberger's property if there is for every sequence (n : n = 1,2,3,…) of open covers of X a sequence (Un : n = 1,2,3,…) such that:

(1) for each n, Un is a member of n, and

(2) {Un: n = 1,2,3,…} is a cover of X.

Rothberger introduced this property in his paper [2]. For convenience we let denote the collection of all open covers of X.

In [3] it was shown that X has Rothberger's property if, and only if, the following partition relation is true for large open covers of X:

This partition relation means:

for every large cover of X, for every coloring

such that for each U ϵ and each large cover there is an i with a large cover of X,

either there is a large cover such that f({A, B}) = 0 whenever {A,B} ϵ ,

or else there is a which is not point–finite such that f{{A, B}) = 1 whenever {A, B} ϵ .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 976 - 980
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Just, W., Miller, A. W., Scheepers, M., and Szeptycki, P. J., Combinatorics of open covers (II), to appear in Topology and its Applications.Google Scholar
[2]Rothberger, F., Eine Verschärfung der Eigenschaft C, Fundamenta Mathematicae, vol. 30 (1938), pp. 50–55.CrossRefGoogle Scholar
[3]Scheepers, M., Combinatorics of open covers (I): Ramsey theory, Topology and its Applications, vol. 69 (1996), pp. 31–62.CrossRefGoogle Scholar