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# Open Problem Book

- Problems from COC2
- Problem 1: Is U
_{fin}(Gamma,Omega )= S_{fin}(Gamma,Omega)?
- Problem 2: Does U
_{fin}(Gamma,Gamma) imply S_{fin}(Gamma,Omega)?
- Problem 3: Does Split(Omega,Omega) imply Split(Lambda,Lambda)?
- 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 U
_{fin}(Gamma,Gamma). Does X then have S_{1}(Gamma,Gamma)?

- Problems from COC3
- Problem 1: Find a space of countable tightness which has property B
_{linear}(Omega_{y},Omega_{y}) but which does not have the property Omega_{y} –> |Omega_{y}|^{2}_{2}
- Problem 2: Find a space of countable tightness which has property C
_{1}(Omega_{y},Omega_{y}) but which does not have the property that for each n and k, Omega_{y} –>(Omega_{y})^{n}_{k}
- Problem 3: Find a space of countable tightness which has property C
_{1}(Omega_{y},Omega_{y}) but which does not have the property Omega_{y} –>[Omega_{y}]^{2}_{3}

- Problems from COC5
- Problem 1: Find a space which satisfies S
_{fin}(D,D), but not S_{fin}(D_{Omega},D_{Omega})
- Problem 2: Is it true that if a space has property S
_{fin}(D_{Omega},D_{Omega}), then each finite power has property S_{fin}(D,D)?
- Problem 3: Is it true that if a space satisfies S
_{fin}(D_{Omega},D_{Omega}), then player ONE has no winning strategy in the game G_{fin}(D_{Omega},D_{Omega})?
- Problem 4: If for space X each element of D
_{Omega} has a countable subset which is in D_{Omega}, then does each finite power of X have countable cellularity?
- Problem 5: Find a space which has property S
_{1}(D,D), but not S_{1}(D_{Omega},D_{Omega})