TY - CHAP

T1 - Menger's and Hurewicz's problems: solutions from ``the book'' and refinements

AU - Tsaban, Boaz

PY - 2011

Y1 - 2011

N2 - We provide simplified solutions of Menger's and Hurewicz's problems and conjectures, concerning generalizations of sigma-compactness. The reader who is new to this field will find a self-contained treatment in Sections 1, 2, and 5. Sections 3 and 4 contain new results, based on the mentioned simplified solutions. The main new result is that there is a set of reals X of cardinality equal to the unbounding number b, and which has the following property: "Given point-cofinite covers U1,U2,... of X, there are for each n sets un,vn in Un, such that each member of X is contained in all but finitely many of the sets u1 union v1,u2 union v2,..." This property is strictly stronger than Hurewicz's covering property, and by a result of Miller and the present author, one cannot prove the same result if we are only allowed to pick one set from each Un.

AB - We provide simplified solutions of Menger's and Hurewicz's problems and conjectures, concerning generalizations of sigma-compactness. The reader who is new to this field will find a self-contained treatment in Sections 1, 2, and 5. Sections 3 and 4 contain new results, based on the mentioned simplified solutions. The main new result is that there is a set of reals X of cardinality equal to the unbounding number b, and which has the following property: "Given point-cofinite covers U1,U2,... of X, there are for each n sets un,vn in Un, such that each member of X is contained in all but finitely many of the sets u1 union v1,u2 union v2,..." This property is strictly stronger than Hurewicz's covering property, and by a result of Miller and the present author, one cannot prove the same result if we are only allowed to pick one set from each Un.

UR - https://www.mendeley.com/catalogue/791c56f3-2a7e-3966-be4b-77161ace84e4/

U2 - 10.1090/conm/533/10509

DO - 10.1090/conm/533/10509

M3 - Chapter

VL - 533

T3 - Contemp. Math.

SP - 211

EP - 226

BT - Set theory and its applications

PB - Amer. Math. Soc., Providence, RI

ER -