  We establish several fundamental properties of analysis-suitable T-splines
which are important for design and analysis. First, we characterize T-spline
spaces and prove that the space of smooth bicubic polynomials, defined over the
extended T-mesh of an analysis-suitable T-spline, is contained in the
corresponding analysis-suitable T-spline space. This is accomplished through
the theory of perturbed analysis-suitable T-spline spaces and a simple
topological dimension formula. Second, we establish the theory of
analysis-suitable local refinement and describe the conditions under which two
analysis-suitable T-spline spaces are nested. Last, we demonstrate that these
results can be used to establish basic approximation results which are critical
for analysis.
