Extending discrete-valued functions

Authors:
John Kulesza, Ronnie Levy and Peter Nyikos

Journal:
Trans. Amer. Math. Soc. **324** (1991), 293-302

MSC:
Primary 54C20

DOI:
https://doi.org/10.1090/S0002-9947-1991-0987164-5

MathSciNet review:
987164

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Abstract: In this paper, we show that for a separable metric space , every continuous function from a subset of into a finite discrete space extends to a continuous function on if and only if every continuous function from into any discrete space extends to a continuous function on . We also show that if there is no inner model having a measurable cardinal, then there is a metric space with a subspace such that every -valued continuous function from extends to a continuous function on all of , but not every discrete-valued continuous function on extends to such a map on . Furthermore, if Martin's Axiom is assumed, such a space can be constructed so that not even -valued functions on need extend. This last result uses a version of the Isbell-Mrowka space having a -embedded infinite discrete subset. On the other hand, assuming the Product Measure Extension Axiom, no such exists.

**[vD]**E. K. van Douwen,*The integers and topology*, Handbook of Set Theoretic Topology, North-Holland, Amsterdam, New York, and Oxford, 1984, pp. 113-167. MR**776622 (87f:54008)****[E]**R. Engelking,*General topology*, Polish Scientific Publishers, Warsaw, 1977. MR**0500780 (58:18316b)****[F]**W. G. Fleissner,*Normal Moore spaces and large cardinals*, Handbook of Set Theoretic Topology, North-Holland, Amsterdam, New York, and Oxford, 1984, pp. 733-760. MR**776635 (86m:54023)****[GJ]**L. Gillman and M. Jerison,*Rings of continuous functions*, Van Nostrand, Princeton, N.J., 1960. MR**0116199 (22:6994)****[K]**K. Kunen,*Set theory: An introduction to independence proofs*, North-Holland, Amsterdam, 1979. MR**756630 (85e:03003)****[N]**K. Nagami,*Dimension theory*, Academic Press, New York, 1970. MR**0271918 (42:6799)****[W]**S. Willard,*General topology*, Addision-Wesley, 1970. MR**0264581 (41:9173)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1991-0987164-5

Keywords:
,
PMEA,
-embedding

Article copyright:
© Copyright 1991
American Mathematical Society