A Topological Characterization of the Real Numbers

S. P. Franklin and G. V. Krishnarao have defined a point x in a topological space (X, 9~) to be a strong cut point if the set X — {x} has two components. They then show that " a connected and locally connected separable and regular space, in which every point is a strong cut point, is homeomorphic to the real line " [1]. We will show that a separable Hausdorjf space (X, ST) is homeomorphic to the real numbers if every xeX is a strong cut point and the set of components of complements of point sets forms a subbasefor the space (X, ^). We shall call a topological space satisfying the conditions of this theorem a cut space or C-space. The subbase we denote by £f and if A £ X then we denote the boundary of A by B(A). We now proceed with a sequence of lemmas.