Spaces of remote points

Given a Tychonoff space X, let ϱ(X) be the set of remote points of X. We view ϱ(X) as a topological space. In this paper we assume that X is metrizable and ask for conditions on Y so that ϱ(X) is homeomorphic to ϱ(Y). This question has been studied before by R.G. Woods and C. Gates. We give some results of the following type: if X has topological property P and ϱ(X) is homeomorphic to ϱ(Y), then Y also has P. We also characterize the remote points of the rationals and irrationals up to some restrictions. Further, we show that ϱ(X) and ϱ(Y) have open dense homeomorphic subspaces if X and Y are both nowhere locally compact, completely metrizable and share the same cellular type, a cardinal invariant we define.