X×Xs

X×Xs denotes the Cartesian product of two mathematical objects, X and Xs. When both are sets, it is the set of all ordered pairs (x, xs) with x in X and xs in Xs. In other contexts, such as topology, algebra, or category theory, the same symbol is used to indicate the product that combines the two objects into a new one that encodes pairs of elements, one from each factor, with structure described by the category in question.

In the setting of sets, the product comes with projection maps π1: X×Xs → X and π2: X×Xs → Xs, given by π1(x, xs) = x and π2(x, xs) = xs. If X and Xs carry additional structure, the product is equipped with the corresponding product structure: a product topology for topological spaces, a direct product structure for groups, rings, or vector spaces, and coordinate-wise operations in the algebraic cases. The universal property characterizes X×Xs as the unique object that receives maps to X and to Xs from any other object in a way that factors through any pair of maps into X and into Xs.

If Xs is a subspace of a larger ambient space, X×Xs can be viewed as a subspace of X×X, obtained by pairing each x with an element of Xs. In analysis and geometry, X×Xs serves as a basic construction for multivariable spaces, graphs of functions (as subsets of X×Xs), and as a building block for more complex objects such as fiber products in algebraic geometry, where products are taken over a base object.

Key properties include |X×Xs| = |X|·|Xs| for finite sets; in topology, the product of compact spaces is compact, and the product of Hausdorff spaces is Hausdorff. For manifolds, the dimension satisfies dim(X×Xs) = dim(X) + dim(Xs).