1995/05/08 by Joan Hart, Hart, Joan, Kenneth Kunen +1
Mathematics · #FOS: Mathematics #Logic (math.LO) #math.LO
paper · pdf · doi:10.48550/arxiv.math/9505204
arxiv created 1995/05/08 · arxiv updated 2016/09/06
Let X be a compact Hausdorff space and M a metric space. E0(X,M) is the set of f in C(X,M) such that there is a dense set of points x in X with f constant 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 beta N - N, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case that M is a Banach space, we discuss the properties of E0(X,M) as a normed linear space. We also build three first countable Eberlein compact spaces, F,G,H, with various E0 properties: For all metric M: E0(F,M) contains only the constant functions, and E0(G,M) = C(G,M). If M is the Hilbert cube or any infinite dimensional Banach space, E0(H,M) is not all of C(H,M), but E0(H,M) = C(H,M) whenever M is a subset of RRn for some finite n.