Locally Constant Functions

Let X be a compact Hausdorff space and M a metric space. E0(X,M) is the set of f ∈ C(X,M) such that there is a dense set of points x ∈ 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 βN\N, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case thatM 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 variousE0 properties. For all metricM , 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) 6= C(H,M), but E0(H,M) = C(H,M) whenever M ⊆ R for some finite n. §0. Introduction. If X is a compact Hausdorff space and M is a metric space, let C(X,M) be the space of all continuous functions from X into M . C(X,M) is a metric space under the sup norm. C(X) denotes C(X,R), which is a (real) Banach algebra. Following [5, 6, 7, 13, 14], if f ∈ C(X,M), let Ωf be the union of all open U ⊆ X such that f is constant on U . Then, E0(X,M) is the set of all f ∈ C(X,M) such that Ωf is dense in X ; these functions are called “locally constant on a dense set”. E0(X) denotes E0(X,R). Clearly, E0(X) is a subalgebra of C(X) and contains all the constant functions. As Bernard and Sidney point out [6, 7, 14], if X is compact metric with no isolated points, then E0(X) is a proper dense subspace of C(X). In this paper, we study the two extreme situations: where E0(X) contains only the constant functions, and where E0(X) = C(X). In §5, we give some justification for studying these two extremes. A standard example of elementary analysis is a monotonic f ∈ C([0, 1]) which does all its growing on a Cantor set; then f is a nonconstant function in E0([0, 1]). More generally, for “many”X , E0(X) separates points inX , and hence (by the Stone-Weierstrass Theorem), is dense in C(X). Specifically, 0.1. Theorem. If X is compact Hausdorff and E0(X) is not dense in C(X), then a. X has a family of 20 disjoint nonempty open subsets. b. X is not locally connected. c. X is not zero-dimensional. 1 Authors supported by NSF Grant DMS-9100665.