Extensions of continuous functions from dense subspaces
HTML articles powered by AMS MathViewer
- by Robert L. Blair PDF
- Proc. Amer. Math. Soc. 54 (1976), 355-359 Request permission
Correction: Proc. Amer. Math. Soc. 106 (1989), 857-858.
Abstract:
Let $X$ and $Y$ be topological spaces, let $S$ be a dense subspace of $X$, and let $f:S \to Y$ be continuous. When $Y$ is the real line ${\mathbf {R}}$, the Lebesgue sets of $f$ are used to provide necessary and sufficient conditions in order that the (bounded) function $f$ have a continuous extension over $X$. These conditions yield the theorem of Taǐmanov (resp. of Engelking and of Blefko and Mrówka) which characterizes extendibility of $f$ for $Y$ compact (resp. realcompact). In addition, an extension theorem of Blefko and Mrówka is sharpened for the case in which $X$ is first countable and $Y$ is a closed subspace of ${\mathbf {R}}$.References
- Robert L. Blair, Filter characterizations of $z$-, $C^*$-, and $C$-embeddings, Fund. Math. 90 (1975/76), no. 3, 285–300. MR 415564, DOI 10.4064/fm-90-3-285-300
- R. Blefko and S. Mrówka, On the extensions of continuous functions from dense subspaces, Proc. Amer. Math. Soc. 17 (1966), 1396–1400. MR 202115, DOI 10.1090/S0002-9939-1966-0202115-X
- Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, N.J., 1952. MR 0050886
- R. Engelking, Remarks on real-compact spaces, Fund. Math. 55 (1964), 303–308. MR 179757, DOI 10.4064/fm-55-3-303-308
- R. Engelking, Outline of general topology, North-Holland Publishing Co., Amsterdam; PWN—Polish Scientific Publishers, Warsaw; Interscience Publishers Division John Wiley & Sons, Inc., New York, 1968. Translated from the Polish by K. Sieklucki. MR 0230273
- 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
- Robert H. McDowell, Extension of functions from dense subspaces, Duke Math. J. 25 (1958), 297–304. MR 97784
- S. Mrówka, On some approximation theorems, Nieuw Arch. Wisk. (3) 16 (1968), 94–111. MR 244938
- S. Mrówka, Characterization of classes of functions by Lebesgue sets, Czechoslovak Math. J. 19(94) (1969), 738–744. MR 248291
- A. D. Taĭmanov, On extension of continuous mappings of topological spaces, Mat. Sbornik N.S. 31(73) (1952), 459–463 (Russian). MR 0050871
- B. Z. Vulih, On the extension of continuous functions in topological spaces, Mat. Sbornik N.S. 30(72) (1952), 167–170 (Russian). MR 0048790
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 54 (1976), 355-359
- DOI: https://doi.org/10.1090/S0002-9939-1976-0390999-0
- MathSciNet review: 0390999