The use of T-splines [30] in Isogeometric Analysis [24] has been proposed in [5] as a tool to enhance the flexibility of isogeometric methods. If T-splines are a very general concept, their success in isogeometric analysis relies upon some basic properties that needs to be true as e.g. (i) linear independence of blending functions and (ii) polynomial reproducibility at element level.In this paper we study these properties for T-splines of a reduced regularity order, namely, for T-splines of degree p and regularity α=p-1.-⌊ p/2⌋ Our results are both for odd and even degree. Under mild assumptions on the underlying T-mesh, T-splines are shown to be linearly independent and the space they span is characterized in terms of piecewise polynomials on a topological extension of the T-mesh. Also, as p is odd, we construct a new topological local refinement algorithm and demonstrate its locality properties through numerical examples. © 2011 Elsevier B.V.