  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 U_1,U_2,... of X, there are for each n sets
u_n,v_n in U_n, such that each member of X is contained in all but finitely
many of the sets u_1 union v_1,u_2 union v_2,..."
  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 U_n.
