Summary: If \(\langle L, < \rangle\) is a dense linear order without end-points and if \(A_1, A_2 \subset\) \(L\) are disjoint dense subsets of \(L\), then \({\mathcal O}_{A_1 A_2}\) denotes the topology on \(L\) generated by the closed intervals \([a_1,a_2]\), where \(a_1\in A_1\) and \(a_2\in A_2\). It is proved that under the Proper Forcing Axiom each two spaces of the form \(\langle {\mathbb R}, {\mathcal O}_{A_1A_2} \rangle\), where \(A_1\) and \(A_2\) are \(\aleph _1\)-dense subsets of the reals, are homeomorphic.