Chebyshev subspaces of finite codimension in spaces of continuous functions

A. L. Brown
1978 Journal of the Australian Mathematical Society  
A. L. Garkavi in 1967 characterized those compact metric spaces X with the property that the space C(X) of real-valued continuous functions possesses Chebyshev subspaces of fine codimension > 2. Here compact Hausdorff spaces with the same property are characterized in terms of certain standard subspaces of the space [0, 1] x {0,1} equipped with a lexicographic order topology. Garkavi's result for metric spaces is exhibited as a corollary. The proof depends upon a simplification of a
more » ... tion by Garkavi of the Chebyshev subspaces of finite codimension in C(X). Subject classification (Amer. Math. Soc. (MOS) 1970): primary 41 A 65, 46 E 15; secondary 54 G 99. A. L. Brown [2] PROOF. Suppose that M is a Chebyshev subspace of C{X) ,so that conditions (a) to (e) are satisfied. It follows from (c) and (d) that s(X) = X. (In particular an isolated point of X has positive A-measure and, therefore, the set A of isolated points is countable.) use, available at https://www.cambridge.org/core/terms. https://doi.
doi:10.1017/s1446788700011575 fatcat:oe5y2nqhbjec7hj45eyogpj5vi