Interpreting modal diamond as the closure of a topological space, we axiomatize the modal logic of each metrizable Stone space and of each extremally disconnected Stone space. As a corollary, we obtain that S4.1 is the modal logic of the Pelczynski compactification of the natural numbers and S4.2 is the modal logic of the Gleason cover of the Cantor space. As another corollary, we obtain an axi...

For every natural number n, any continuous function on the product of X1×X2×...×Xn pseudocompact spaces extends to a separately βX1×βX2×...×βXn their Stone-Cech compactifications.

We prove that the following statement follows from the Open Colouring Axiom (OCA): if X is locally compact σ-compact but not compact and if itš Cech-Stone remainder X * is a continuous image of ω * then X is the union of ω and a compact set. It follows that the remainders of familiar spaces like the real line or the sum of countably many Cantor sets need not be continuous images of ω * .

An n x m rational matrix A is said to be partition regular if for every finite colouring oi \ there is a monochromatic vector x s N with Ax = 0. A set D c f̂J is said to be partition regular for A (or for the system of equations Ax = 0) if for every finite colouring of D there is a monochromatic x e D' with Ax = 6. In this paper we show that for every n there is a set that is partition regular f...

The aim of this article is to obtain a characterization of the canonical extension of Boolean homomorphisms via the Stone-Čech compactification.