Equivalences of Rothberger’s property in finite powers Marion Scheepers

S1(A,B)S1(A,B) For each sequence (A n :n=1,2,3,…) of elements of A, there is a sequence (B n :n=1,2,3,…) such that: 1.For each n, B n  A n, and 2.{B n :n=1,2,3,…}  B.

Fix A:={a 1,a 2,…,a n,…} in A. s B C The Ellentuck topology. [s,B] = {C : s  C  s  B} For B  X and s  A finite: s < B: a n  s and a m  B  n<m. X:= [A]  := Ellentuck(A)

For each countable A in A, and each R  Ellentuck(A): E(A,B) Ellentuck(A) Ellentuck(A)  B R (1) R has the Baire property in Ellentuck(A)  B. (2) For each B  A with B  A, and each finite s  A, there is an infinite C  B|s with: (a) C  B, and (b) either ([s,C]  B)  R, or else [s,C]  B  R = . E(A,B) ≡ (1)  (2) Ellentuck’s Theorem: E([N] ,[N]  )

(A,B)-Mathias reals A  A (s,B)  M (A,B) (A) if: (i) s  A finite; (ii) B  A and B  B and (iii) s < B. s B B1B1 s1s1 (s,B) < (s 1, B 1 ) if: (i) s  s 1 ; (ii) B 1  B; (iii) s 1 \s  B A

Theorem The following are equivalent: 2) E( ,  ) 1) S 1 ( ,  ) 3) For each (s,B) in M ( ,  ) (A), and each sentence  in the forcing language of M  (A) there is a C  B with C   such that (s,C) ╟  or (s,C) ╟ ¬ 

E( ,  ) FG( ,  ) NW( ,  )  ()nk ()nk S 1 ( ,  ) M-Forcing( ,  ) One ↯ G 1 ( ,  )

Examples of A and B O All open covers of a given space  All large open covers of a given space  All omega covers K All k-covers O fd All FD-covers O cfd All CFD-covers O kfd All kFD-covers  o D  Various groupability properties