Locally constant functions

摘要

Let X be a compact Hausdorff space and M a metric space. E0(X,M) is the set of / e G(X,M) such that there is a dense set of points xe X with_/ constan on some neighborhood of x. We describe some general classes of X for which E0{X,M) is all of C(X, M). These include /3NN, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case where M is a Banach space, we discuss the properties of E0(X,M) as a normed linear space. We also bmld three first countable Eberlein compact spaces, F, G, H, with various f 0 Properties _For all metric M, EQ (F, M) contains only the constant functions, and E0(G,M) - W*?)-ll is the Hilbert cube or any infinite-dimensional Banach space, then E0{H,M) ? O,mh but E0(H, M) = C(H, M) whenever M C Rn for some finite n.