Covering an uncountable square by countably many continuous functions

We prove that there exists a countable family of continuous real functions whose graphs together with their inverses cover an uncountable square, i.e. a set of the form $X\times X$, where $X$ is an uncountable subset of the real line. This extends Sierpi\'nski's theorem from 1919, saying that $S\times S$ can be covered by countably many graphs of functions and inverses of functions if and only if the size of $S$ does not exceed $\aleph_1$. Our result is also motivated by Shelah's study of planar Borel sets without perfect rectangles.Comment: Added new results (9 pages

Linearly ordered compacta and Banach spaces with a projectional resolution of the identity

We construct a compact linearly ordered space $K$ of weight aleph one, such that the space $C(K)$ is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, $K$ is a continuous image of a Valdivia compact and every separable subspace of $C(K)$ is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.Comment: 13 page