Extending discrete-valued functions

Kulesza, John; Levy, Ronnie; Nyikos, Peter

doi:10.1090/S0002-9947-1991-0987164-5

Extending discrete-valued functions
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by John Kulesza, Ronnie Levy and Peter Nyikos PDF

Trans. Amer. Math. Soc. 324 (1991), 293-302 Request permission

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.

References

Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622
Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
William G. Fleissner, The normal Moore space conjecture and large cardinals, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 733–760. MR 776635
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1983. An introduction to independence proofs; Reprint of the 1980 original. MR 756630
Keiô Nagami, Dimension theory, Pure and Applied Mathematics, Vol. 37, Academic Press, New York-London, 1970. With an appendix by Yukihiro Kodama. MR 0271918
Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581