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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Extending discrete-valued functions


Authors: John Kulesza, Ronnie Levy and Peter Nyikos
Journal: Trans. Amer. Math. Soc. 324 (1991), 293-302
MSC: Primary 54C20
MathSciNet review: 987164
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Abstract: In this paper, we show that for a separable metric space $ X$, every continuous function from a subset $ S$ of $ X$ into a finite discrete space extends to a continuous function on $ X$ if and only if every continuous function from $ S$ into any discrete space extends to a continuous function on $ X$. We also show that if there is no inner model having a measurable cardinal, then there is a metric space $ X$ with a subspace $ S$ such that every $ 2$-valued continuous function from $ S$ extends to a continuous function on all of $ X$, but not every discrete-valued continuous function on $ S$ extends to such a map on $ X$. Furthermore, if Martin's Axiom is assumed, such a space can be constructed so that not even $ \omega$-valued functions on $ S$ need extend. This last result uses a version of the Isbell-Mrowka space $ \Psi$ having a $ {C^ * }$-embedded infinite discrete subset. On the other hand, assuming the Product Measure Extension Axiom, no such $ \Psi$ exists.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-0987164-5
PII: S 0002-9947(1991)0987164-5
Keywords: $ \Psi$, PMEA, $ \alpha$-embedding
Article copyright: © Copyright 1991 American Mathematical Society