  We analyze the space of bivariate functions that are piecewise polynomial of
bi-degree \textless{}= (m, m') and of smoothness r along the interior edges of
a planar T-mesh. We give new combinatorial lower and upper bounds for the
dimension of this space by exploiting homological techniques. We relate this
dimension to the weight of the maximal interior segments of the T-mesh, defined
for an ordering of these maximal interior segments. We show that the lower and
upper bounds coincide, for high enough degrees or for hierarchical T-meshes
which are enough regular. We give a rule of subdivision to construct
hierarchical T-meshes for which these lower and upper bounds coincide. Finally,
we illustrate these results by analyzing spline spaces of small degrees and
smoothness.
