Dispersed images of topological spaces and uncomplemented subspaces of $C(X)$
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- by John Warren Baker PDF
- Proc. Amer. Math. Soc. 41 (1973), 309-314 Request permission
Abstract:
Let $\Gamma (\alpha )$ denote the set of ordinals not exceeding $\alpha$ with its interval topology. We show that, if $X$ is a $0$-dimensional Hausdorff space and $\alpha$ is a denumerable ordinal such that the $\alpha$th derived set of $X$ contains $n$ points where $n < \omega$, there is a map of $X$ onto $\Gamma ({\omega ^\alpha } \cdot n)$. Maps of completely regular spaces into the unit interval are considered and a noncompact analogue of a theorem of Pełczyński and Semadeni is obtained. Finally, these results are used to give a simple proof to the following theorem: If $X$ is completely regular and ${X^{(\omega )}} \ne \emptyset$, there is an uncomplemented subspace $H$ of $C(X)$ which is isometrically isomorphic to $C(Y)$ for some compact metric space $Y$.References
- D. Amir, Continuous functions’ spaces with the bounded extension property, Bull. Res. Council Israel Sect. F 10F (1962), 133–138 (1962). MR 143026
- Richard Arens, Projections on continuous function spaces, Duke Math. J. 32 (1965), 469–478. MR 181882
- John Warren Baker, Some uncomplemented subspaces of $C(X)$ of the type $C(Y)$, Studia Math. 36 (1970), 85–103. MR 275356, DOI 10.4064/sm-36-2-85-103
- John Warren Baker, Compact spaces homeomorphic to a ray of ordinals, Fund. Math. 76 (1972), no. 1, 19–27. MR 307197, DOI 10.4064/fm-76-1-19-27
- John Warren Baker, Ordinal subspaces of topological spaces, General Topology and Appl. 3 (1973), 85–91. MR 324623 —, Uncomplemented $C(X)$-subalgebras of $C(X)$, Trans. Amer. Math. Soc. (to appear).
- Seymour Z. Ditor, Averaging operators in $C(S)$ and lower semicontinuous sections of continuous maps, Trans. Amer. Math. Soc. 175 (1973), 195–208. MR 312228, DOI 10.1090/S0002-9947-1973-0312228-8
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- 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
- Casimir Kuratowski, Topologie. Vol. II, Monografie Matematyczne, Tom 21, Państwowe Wydawnictwo Naukowe, Warsaw, 1961 (French). Troisième édition, corrigèe et complétée de deux appendices. MR 0133124
- A. Pełczyński and Z. Semadeni, Spaces of continuous functions. III. Spaces $C(\Omega )$ for $\Omega$ without perfect subsets, Studia Math. 18 (1959), 211–222. MR 107806, DOI 10.4064/sm-18-2-211-222
- Z. Semadeni, Sur les ensembles clairsemés, Rozprawy Mat. 19 (1959), 39 pp. (1959) (French). MR 107849 W. Sierpiński, Cardinal and ordinal numbers, 2nd rev. ed., Monografie Mat., Vol. 34, PWN, Warsaw, 1965. MR 33 #2549.
- R. Telgársky, Total paracompactness and paracompact dispersed spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 567–572 (English, with Russian summary). MR 235517
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 41 (1973), 309-314
- MSC: Primary 54C05; Secondary 46E15
- DOI: https://doi.org/10.1090/S0002-9939-1973-0320984-3
- MathSciNet review: 0320984