WEAK CHEBYSHEV SPACES ON LOCALLY ORDERED TOPOLOGY SPACE AND THE RELATED CONTINUOUS METRIC SELECTIONS

Let C(X)be the space of all continuous real-valued functions on a compact Hausdorffspace X under the uniform norm:‖f‖=max{|f(x)|:x∈X}.For G(?)C(X),defineP_G(f)={g∈G:‖f-g‖=inf{‖f-p‖:p∈G}}.If there exists a continuous mapping S from C(X)to G such that S(f)∈P_G(f)for everyf 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 vanishon any open subset of X.In this paper,the author gives several characterizations of Z-subspaces G whose metricprojections 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 hasa continuous selection if and only if every nonzero g in G has at most n zeros and has atmost n-1 zeros with sign changes.