R1×m

R1×m is a notation used to denote the Cartesian product of the real line with a finite index set of size m, commonly written as R × {1, 2, ..., m}. It is the product topology on the space consisting of pairs (x, i) where x ∈ R and i ∈ {1,...,m}.

Topologically, R × {1,...,m} is the disjoint union of m copies of the real line. Each component {i} × R is homeomorphic to R, so the whole space has m connected components. If m = 1, the space is simply the real line. For m > 1, the space is not connected but remains a 1-dimensional manifold in each component.

Properties and structure: the space is locally Euclidean of dimension 1, second countable, locally compact, and Hausdorff. It is metrizable; a convenient metric comes from the product topology, for example d((x,i),(y,j)) = sqrt((x−y)^2 + δ_{i≠j}), where δ_{i≠j} is 0 if i = j and 1 otherwise. Different equivalent metrics can be used, provided they generate the same topology. The space is complete with respect to standard product metrics and is a simple example of a product manifold composed of multiple copies of R.

Uses and interpretations: R × {1,...,m} serves as a model for systems with m independent channels or regimes, with each index i labeling a channel. Function spaces on this product decompose into m-tuples of functions on R, i.e., maps f: R × {1,...,m} → S correspond to (f_1,...,f_m) with f_i: R → S. In differential geometry and analysis, it illustrates the concept of a finite disjoint union of a single manifold.