We will address local refinement of a tensor product grid specified as a sequence of inserted line segments parallel to the knot lines. The line segments are assigned multiplicities to model the continuity across each line segments individually. We obtain a quadrilateral grid with T-junctions in the parameter domain, and a collection of tensor product B-splines on this mesh here named an LR-mesh. The approach applies equally well in dimensions higher than two. By refining according to a hand-in-hand principle between the dimension of the spline space over the LR-mesh, the spline space spanned by the Locally Refined B-splines and the number of locally refined B-splines the LR B-splines are linear independent and form a basis. Specifically we will for the 2-variate case address how this applies to the insertion of a new line segment, to the extension of an existing line segment and to the merging of two separate lines segments.