Some notes on the superintuitionistic logic of chequered subsets of ${\mathbb{R}}^{\mathrm{\infty }}$

I investigate the superintuitionistic analogue of the modal logic of chequered subsets of ${\mathbb{R}}^{\mathrm{\infty }}$ introduced by van Benthem et al. It is observed that this logic possesses the disjunction property, contains the Scott axiom, fails to contain the Kreisel-Putnam axiom and it is a sublogic of the Medvedev logic.