Paracompactness, normality, and related properties of topologies on infinite products

There are a number of ways to create products of spaces, where we say that a topology on Πi∈IXiΠi∈IXi is a product (as opposed to the product) topology iff each projection of an open set is open. The smallest such product topology is the Tychonoff product (a.k.a. the product): a basic set restricts on finitely many coordinates. The largest such product topology is the box product: a basic set restricts independently on each coordinate. In this paper we will focus on the box product (which Mary Ellen Rudin worked on and proselytized about), and on the uniform box product (a naturally defined intermediate product between the Tychonoff product and the box product).The first section will discuss the box product, focusing on Rudin's work and techniques which derive (although not necessarily consciously) from her work, ending with open questions. The second section of the paper will discuss the uniform box product.