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1. Notation and terminology. LetX = (X,T ) be a topological space. We shall denote topological spaces by boldface letters and underlying sets by lightface letters. X is said to be compact iff every open cover U of X has a finite subcover V. Equivalently, X is compact iff every family G of closed subsets of X with the finite intersection property, fip for abbreviation, has a non-empty intersection. A subset A of X is called a regular open set if A = int(A). It is known that the set of all regular open sets of X forms a Boolean algebra under the following set of operations:

On BPI Restricted to Boolean Algebras of Size Continuum