CH, a problem of Rolewicz and bidiscrete systems

TL;DR

This paper constructs a special non-metrizable compact space under CH with unique properties affecting semi-biorthogonal sequences in its continuous function space, and introduces bidiscrete systems in compact spaces.

Contribution

It provides a construction of a non-metrizable compact space with specific properties and introduces the concept of bidiscrete systems in compact spaces.

Findings

Any uncountable semi-biorthogonal sequence in C(K) is of a specific kind.

Every infinite compact space has a bidiscrete system of size equal to its density.

C(K) has a biorthogonal system of size equal to the space's density.

Abstract

We give a construction under $CH$ of a non-metrizable compact Hausdorff space $K$ such that any uncountable 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 Lindel\"of 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. We introduce the notion of bidiscrete systems in compact spaces and note that every infinite compact Hausdorff space $K$ must have a bidiscrete system of size $d(K)$, the density of $K$. This, in particular, implies that $C(K)$ has a biorthogonal system of size $d(K)$.