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Recall that a P-set is a closed set X such that the intersection of countably many neighborhoods of X is again a neighborhood of X. We show that if 𝔱 = 𝔠 then there is a minimal right ideal of (βℕ,+) that is also a P-set. We also show that the existence of such P-sets implies the existence of P-points; in particular, it is consistent with ZFC that no minimal right ideal is a P-set. As an application of these results, we prove that it is both consistent with and independent of ZFC that the shift map and its inverse are (up to isomorphism) the unique chain transitive autohomeomorphisms of ℕ*.
W. R. Brian. "P-sets and minimal right ideals in ℕ*." Fundamenta Mathematicae 229.3 (2015): 277-293. <http://eudml.org/doc/286651>.
@article{W2015, abstract = {Recall that a P-set is a closed set X such that the intersection of countably many neighborhoods of X is again a neighborhood of X. We show that if 𝔱 = 𝔠 then there is a minimal right ideal of (βℕ,+) that is also a P-set. We also show that the existence of such P-sets implies the existence of P-points; in particular, it is consistent with ZFC that no minimal right ideal is a P-set. As an application of these results, we prove that it is both consistent with and independent of ZFC that the shift map and its inverse are (up to isomorphism) the unique chain transitive autohomeomorphisms of ℕ*.}, author = {W. R. Brian}, journal = {Fundamenta Mathematicae}, keywords = {topology/dynamics/algebra in and ; -sets; (minimal) right ideals; tower number; thick sets; chain transitivity}, language = {eng}, number = {3}, pages = {277-293}, title = {P-sets and minimal right ideals in ℕ*}, url = {http://eudml.org/doc/286651}, volume = {229}, year = {2015}, }
TY - JOUR AU - W. R. Brian TI - P-sets and minimal right ideals in ℕ* JO - Fundamenta Mathematicae PY - 2015 VL - 229 IS - 3 SP - 277 EP - 293 AB - Recall that a P-set is a closed set X such that the intersection of countably many neighborhoods of X is again a neighborhood of X. We show that if 𝔱 = 𝔠 then there is a minimal right ideal of (βℕ,+) that is also a P-set. We also show that the existence of such P-sets implies the existence of P-points; in particular, it is consistent with ZFC that no minimal right ideal is a P-set. As an application of these results, we prove that it is both consistent with and independent of ZFC that the shift map and its inverse are (up to isomorphism) the unique chain transitive autohomeomorphisms of ℕ*. LA - eng KW - topology/dynamics/algebra in and ; -sets; (minimal) right ideals; tower number; thick sets; chain transitivity UR - http://eudml.org/doc/286651 ER -