A topological characterization of sets of real numbers
Mary Ellen Rudin · Pacific Journal of Mathematics · 1957
We will say that a space E is of class L if E is a separable metric space which satisfies the following conditions :(1) Each component ofEis a point or an arc (closed, open, or halfopen), and no interior point of an arc-component A is a limit point of E-A.(2) Each point of E has arbitrarily small neighborhoods whose boundaries are finite sets.The purpose of this note is to show that a necessary and sufficient condition that a space be homeomorphic to a set of real numbers is that it be of class L.This gives an affirmative answer to a question raised by de Groot in [1].In [2] L. W. Cohen proved that a separable metric space is homeomorphic to a set of real numbers if and only if it satisfies (1) above and (3) and (4) below :(3) E is zero-dimensional at each of its point-components.(4) If p is an end point of an arc-component A, then the space (E-A)\J {p} is zero-dimensional at p.Any set of real numbers is clearly of class L. To prove the converse it is sufficient to show that every space of class L satisfies conditions (3) and (4).To this end it is clearly enough to show the following :If X is a component of the space E of class L and ε is a positive number, there is a set U(X, ε) which is both open and closed, contains X, and is contained in the union of X with the ^-neighborhoods of its endpoints (if any).Suppose X is a component of a space E of class L and e is a positive number.There exists an open set V which contains X but contains no point whose distance from X exceeds e, such that the boundary B of V is finite if X is a point, we can apply (2) directly to obtain V if X is an arc, let V consist of X plus type (2) neighborhoods of the end points of X (if any).Let G denote the sets of all points p of E such that E is the union of two mutually separated sets S 9 and T p , where S p contains X and T p contains p.