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A consequence of the Proper Forcing Axiom in topology. (English) Zbl 1047.03038

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
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