and only if for every proper filter over the intersection of all the complexes contained in the filter is non-empty. These definitions arise from considering the topology generated by the complexes of a field of sets. (It is just one of notable topologies on the given set of points; it often happens that another topology is given, with quite different properties, in particular, not zero-dimensional). Given a field of sets the complexes form a base for a topology. We denote by the corresponding topological space, where is the topology formed by taking arbitrary unions of complexes. Then is always a zero-dimensional space. is a Hausdorff space if and only if is separative. is a compact space with compact open sets if and only if is compact. is a Boolean space with clopen sets if and only if is both separative and compact (in which case it is described as being descriptive) The Stone representation of a Boolean algebra is always separative and compact; the corresponding Boolean space is known as the Stone space of the Boolean algebra. The clopen sets of the Stone space are then precisely the complexes of the Stone representation. The area of mathematics known as Stone duality is founded on the fact that the Stone representation of a Boolean algebra can be recovered purely from the corresponding Stone space whence a duality exists between Boolean algebras and Boolean spaces. Fields of sets with additional structure Sigma algebras and measure spaces If an algebra over a set is closed under countable unions (hence also under countable intersections), it is called a sigma algebra and the corresponding field of sets is called a measurable space. The complexes of a measurable space are called measurable sets. The Loomis-Sikorski theorem provides a Stone-type duality between countably complete Boolean algebras (which may be called abstract sigma algebras) and measurable spaces. A measure space is a triple where is a measurable space and is a measure defined on it. If is in fact a probability measure we speak of a probability space and call its underlying measurable space a sample space. The points of a sample space are called samples and represent potential outcomes while the measurable sets (complexes) are called events and represent properties of outcomes for which we wish to assign probabilities. (Many use the term sample space simply for the underlying set of a probability space, particularly in the