# Open Problem Book

• Problems from COC2
1. Problem 1: Is Ufin(Gamma,Omega )= Sfin(Gamma,Omega)?
2. Problem 2: Does Ufin(Gamma,Gamma) imply Sfin(Gamma,Omega)?
3. Problem 3: Does Split(Omega,Omega) imply Split(Lambda,Lambda)?
4. Problem 4: Let X be a set of real numbers which does not contain a perfect set of real numbers but which has the Hurewicz property Ufin(Gamma,Gamma). Does X then have S1(Gamma,Gamma)?
• Problems from COC3
1. Problem 1: Find a space of countable tightness which has property Blinear(Omegay,Omegay) but which does not have the property Omegay –> |Omegay|22
2. Problem 2: Find a space of countable tightness which has property C1(Omegay,Omegay) but which does not have the property that for each n and k, Omegay –>(Omegay)nk
3. Problem 3: Find a space of countable tightness which has property C1(Omegay,Omegay) but which does not have the property Omegay –>[Omegay]23
• Problems from COC5
1. Problem 1: Find a space which satisfies Sfin(D,D), but not Sfin(DOmega,DOmega)
2. Problem 2: Is it true that if a space has property Sfin(DOmega,DOmega), then each finite power has property Sfin(D,D)?
3. Problem 3: Is it true that if a space satisfies Sfin(DOmega,DOmega), then player ONE has no winning strategy in the game Gfin(DOmega,DOmega)?
4. Problem 4: If for space X each element of DOmega has a countable subset which is in DOmega, then does each finite power of X have countable cellularity?
5. Problem 5: Find a space which has property S1(D,D), but not S1(DOmega,DOmega)
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