Homeomorphisms of Two-point Sets
نویسنده
چکیده
Given a cardinal κ ≤ c, a subset of the plane is said to be a κ-point set if and only it it meets every line in precisely κ many points. In response to a question of Cobb, we show that for all 2 ≤ κ, λ < c there exists a κ-point set which is homeomorphic to a λ-point set, and further, we also show that it is consistent with ZFC that for all 2 ≤ κ < c, there exists a κ-point set X such that for all 2 ≤ λ < c, X is homeomorphic to a λ-point set. On the other hand, we prove that is is consistent with ZFC that for all 2 ≤ κ < c there exists a κ-point set X which is not homeomorphic to a λ-point set for any distinct 2 < λ ≤ c.
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