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Register a new userThe Strength of Mengers Conjecture
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Menger conjectured that subsets of R with the Menger property must be ${sigma}$-compact. While this is false when there is no restriction on the subsets of R, for projective subsets it is known to follow from the Axiom of Projective Determinacy, which has considerable large cardinal consistency strength. We note that in fact, Mengers conjecture for projective sets has consistency strength of only an inaccessible cardinal.
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Menger conjectured that subsets of $mathbb R$ with the Menger property must be $sigma$-compact. While this is false when there is no restriction on the subsets of $mathbb R$, for projective subsets it is known to follow from the Axiom of Projective Determinacy, which has considerable large cardinal consistency strength. We show that the perfect set version of the Open Graph Axiom for projective sets of reals, with consistency strength only an inaccessible cardinal, also implies Mengers conjecture restricted to this family of subsets of $mathbb R$.
We prove that every usco multimap $Phi:Xto Y$ from a metrizable separable space $X$ to a GO-space $Y$ has an $F_sigma$-measurable selection. On the other hand, for the split interval $ddot{mathbb I}$ and the projection $P:ddot{mathbb I}^2to{mathbb I}^2$ of its square onto the unit square ${mathbb I}^2$, the usco multimap $P^{-1}:{mathbb I}^2multimapddot{mathbb I}^2$ has a Borel ($F_sigma$-measurable) selection if and only if the Continuum Hypothesis holds. This CH-example shows that know results on Borel selections of usco maps into fragmentable compact spaces cannot be extended to a wider class of compact spaces.
We provide conceptual proofs of the two most fundamental theorems concerning topological games and open covers: Hurewiczs Theorem concerning the Menger game, and Pawlikowskis Theorem concerning the Rothberger game.
We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of omega_1 is hereditarily paracompact.
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