CH, a problem of Rolewicz and bidiscrete systems

We give a construction under CH of a non-metrizable compact Hausdorff space K such that any uncountable ‘nice’ semi-biorthogonal sequence in C ( K ) must be of a very specific kind. The space K has many nice properties, such as being hereditarily separable, hereditarily Lindelöf and a 2-to-1 continuous preimage of a metric space, and all Radon measures on K are separable. However K is not a Rosenthal compactum. roduce the notion of a bidiscrete system in a compact space K. These are subsets of K 2 which determine biorthogonal systems of a special kind in C ( K ) that we call nice. We note that for every infinite compact Hausdorff space K, the space C ( K ) has a bidiscrete system and hence a nice biorthogonal system of size d ( K ) , the density of K.