Cleavability in ordered spaces (Q5947334)

A generalized ordered space (GO-space) is a set \(X\) with a linear ordering \(<\) and a Hausdorff topology \({\mathcal T}\) that has a base of order-convex sets. Let \({\mathcal M}\) be the class of metrizable spaces, and let \({\mathcal S}\) be the class of separable metrizable spaces. A space \(X\) is cleavable over \({\mathcal M}\) if for every subset \(A\subseteq X\) there is a space \(M_A\in{\mathcal M}\) and a continuous function \(f_A:X\to M_A\) such that if \(x\in A\) and \(y\in X-A\), then \(f_A(x)\neq f_A(y)\). A space \(X\) is absolutely cleavable over \({\mathcal M}\) if there is a continuous 1-1 mapping from \(X\) into a member of \({\mathcal M}\). The terms cleavable over \({\mathcal S}\) and absolutely cleavable over \({\mathcal S}\) are defined analogously. A topological space is divisible by cozero sets if for each \(A\subseteq X\) there is a countable collection \({\mathcal C}\) of cozero sets of continuous real-valued functions such that if \(x\in A\) and \(y\in X-A\), then there is \(C\in{\mathcal C}\) with \(x\in C\) and \(y\not\in C\). The authors prove a number of characterizations of these concepts. The following are representative. Theorem. A GO-space is absolutely cleavable over \({\mathcal M}\) if and only if it is cleavable over \({\mathcal M}\). Theorem. A topological space is cleavable over \({\mathcal S}\) if and only if it is divisible by cozero sets. Theorem. The following are equivalent for a GO-space \(X\): (i) \(X\) is absolutely cleavable over \({\mathcal S}\); (ii) \(X\) is cleavable over \({\mathcal S}\) and \(c(X)\leq{\mathfrak c}\); (iii) \(X\) is divisible by cozero sets and \(c(X)\leq{\mathfrak c}\). The authors also give a number of examples and open problems.