Kurilić, M. S.; Pavlović, A. A consequence of the Proper Forcing Axiom in topology. (English) Zbl 1047.03038 Publ. Math. Debr. 64, No. 1-2, 15-20 (2004). 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. MSC: 03E35 Consistency and independence results 03E65 Other set-theoretic hypotheses and axioms 54A35 Consistency and independence results in general topology 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 03E75 Applications of set theory Keywords:proper forcing axiom; real line; closed intervals; linear orders PDFBibTeX XMLCite \textit{M. S. Kurilić} and \textit{A. Pavlović}, Publ. Math. Debr. 64, No. 1--2, 15--20 (2004; Zbl 1047.03038)