RN×P

RN×P refers to the Cartesian product of R^N and P, where R^N denotes the N-dimensional Euclidean space and P is a set equipped with a topology (or an associated metric). The elements are ordered pairs (x, p) with x ∈ R^N and p ∈ P. When R^N and P are endowed with their respective topologies, RN×P carries the product topology, whose open sets are unions of products U × V with U open in R^N and V open in P. If P is a metric space, RN×P can be metrized by standard product metrics, such as d((x,p),(y,q)) = max(||x−y||, d_P(p,q)). Projections π_R: (x,p) ↦ x and π_P: (x,p) ↦ p are continuous.

Subsets of RN×P are products A×B with A⊆R^N and B⊆P; basic properties include that A×B is open in RN×P if A and B are open, and that RN×P inherits properties from its factors, like connectedness or compactness, under appropriate conditions (R^N is locally compact, etc.). The space is widely used in analysis and topology to model pairs of data, parameters, or states, e.g., a real-valued vector parameter together with a discrete label.

Common special cases include fixing P to a singleton, yielding RN×{p0} ≅ R^N, or letting P be a finite set to obtain a finite disjoint union of copies of R^N. See also Cartesian product, product topology, Euclidean space, and product metric.