A motivating example is that the product of manifolds is a manifold in a fairly natural way, and the dimension of the result is the sum of the dimensions of the

A motivating example is that the product of manifolds is a manifold in a fairly natural way, and the dimension of the result is the sum of the dimensions of the factors. The factors come in one of two flavors, compact and not compact, with compact roughly meaning finite volume, with the caveat that the topological analog to volume is a little strange and only gives two answers, infinite and finite. The coordinates of a product of a compact and a non compact manifold are then either constrained in a compact space or not. Not every manifold is a product in this way though, for an easy example, theres the mobius band (with the boundary removed). This is a "twisted product" of a compact manifold and a non compact manifold, so we have some notion of one small and one large dimension here. even though we dont have explicit projections onto each factor.