R×X0

R×X0 denotes the Cartesian product of the set R of real numbers and a set X0. It consists of all ordered pairs (r, x0) where r ∈ R and x0 ∈ X0. The notation is standard in mathematics for forming a product space from two components that may carry additional structure.

If X0 carries extra structure, the product inherits a corresponding product structure. For example, if X0 is a topological space, R×X0 is equipped with the product topology. If X0 has a measure, the product space carries the product sigma-algebra. If X0 is a vector space, R×X0 can be treated as a direct product of vector spaces, with coordinate-wise addition and scalar multiplication.

In metric and geometric contexts, the product space can be given a metric such as d((r1,x1),(r2,x2)) = max(|r1−r2|, dX(x1,x2)) when X0 is a metric space with metric dX. If X0 is finite with n elements, R×X0 is a union of n copies of R, and its cardinality is that of R; more generally, |R×X0| = |R| × |X0|.

Special cases illustrate common identifications: if X0 ≅ R^k, then R×X0 ≅ R^{k+1}; if X0 is a finite set, the product behaves as several real-valued layers indexed by X0.

R×X0 is widely used to model joint data where one component is real-valued and the other comes from X0, to parameter spaces in analysis, and to describe product spaces in geometry and applied disciplines.