Coherent omission of intervals: Menger's and Hurewicz's problems

Sun, 15/03/2015 - 14:05

We introduce Menger and Hurewicz covering properties, which are generalizations of sigma-compactness. Menger and Hurewicz conjectured that, for subsets of the real line, the above properties were equivalent to sigma-compactness. Using topological and an elegant combinatorial method (coherent omission of intervals), we show (in ZFC) that they are false. We consider also stronger covering properties, relations between them and we give examples of such sets of reals. After that we obtain the solution to the Hurewicz problem: Is there in ZFC an example of set of reals which is Menger but not Hurewicz? Finally we show some results concerning behavior of Menger and Hurewicz properties in finite products.


The methods, proofs, and results, are mainly due to Tsaban and his collaborators. The last lecture will include new results, due to Tsaban and the speaker.