In this paper, we define the topological structures for an arbitrary axis-aligned box partition of a parametric d-dimensional box-shaped limited domain in R^d. Then we define the d-variate spline space over this partition with given polynomial degrees and arbitrary continuity constraints. We then use homological techniques to show that the dimension of this spline space can be split up as dim S(N) = C + H, where the first term is a combinatorial easily calculated term that only depends on the topological structure, polynomial degrees and continuity constraints, while the second term is an alternating sum of dimensions of homological terms. They are often zero, but not always, and might even in some special situations depend on the parameterization. We give explicit expressions for the terms in tensor product spaces, before we look at how the homology modules are tied together during a refinement process. Eventually we discuss the cases d=2 and d=3.