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| Weak Chebyshev Spaces on Locally Ordered Topology Space and the RelatedContinuous Metric Selections |
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Citation： |
Li Wu.Weak Chebyshev Spaces on Locally Ordered Topology Space and the RelatedContinuous Metric Selections[J].Chinese Annals of Mathematics B,1987,8(4):420~427 |
| Page view： 818
Net amount： 694 |
Authors： |
Li Wu; |
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| Abstract： |
Let C(X) be the space of all continuous real-valued functions on a compact Hausdorff space X under the uniform norm:
||f||=max{|f(x)|:x\in X}.
For G\subseteqq C(X), define
P_G(f)={g\in G:||f-g||=inf{||f-p||:p\in G}}.
If there exists a continuous mapping 8 from C(X) to G such that S(f)\in P_G(f) for every f in C(X), then S is called a continuous selection of the metric projection P_G.
And G is called a Z-subspace of C(X), if, for every nonzero g in G, g does not vanish on any open subset of X.
In this paper, the author gives several characterizations of Z-subspaces G whose metric projections P_G have continuous selections. The following results are obtained:
If X is locally connected and G is an n-dimensional Z-subspace of C(X), then P_G has a continuous selection if and only if every nonzero g in G has at most n zeros and has at most n—1 zeros with sign changes. |
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