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Assuming the continuum hypothesis there is an inseparable sequence of length ω₁ that contains no Lusin subsequence, while if Martin's Axiom and ¬ CH are assumed then every inseparable sequence (of length ω₁) is a union of countably many Lusin subsequences.
Uri Abraham, and Saharon Shelah. "Lusin sequences under CH and under Martin's Axiom." Fundamenta Mathematicae 169.2 (2001): 97-103. <http://eudml.org/doc/281882>.
@article{UriAbraham2001, abstract = {Assuming the continuum hypothesis there is an inseparable sequence of length ω₁ that contains no Lusin subsequence, while if Martin's Axiom and ¬ CH are assumed then every inseparable sequence (of length ω₁) is a union of countably many Lusin subsequences.}, author = {Uri Abraham, Saharon Shelah}, journal = {Fundamenta Mathematicae}, keywords = {Lusin sequence; Martin's axiom; continuum hypothesis}, language = {eng}, number = {2}, pages = {97-103}, title = {Lusin sequences under CH and under Martin's Axiom}, url = {http://eudml.org/doc/281882}, volume = {169}, year = {2001}, }
TY - JOUR AU - Uri Abraham AU - Saharon Shelah TI - Lusin sequences under CH and under Martin's Axiom JO - Fundamenta Mathematicae PY - 2001 VL - 169 IS - 2 SP - 97 EP - 103 AB - Assuming the continuum hypothesis there is an inseparable sequence of length ω₁ that contains no Lusin subsequence, while if Martin's Axiom and ¬ CH are assumed then every inseparable sequence (of length ω₁) is a union of countably many Lusin subsequences. LA - eng KW - Lusin sequence; Martin's axiom; continuum hypothesis UR - http://eudml.org/doc/281882 ER -