Topographically convex regions within a catchment basin represent varied degrees of hill-slopes. The non-network space (M), the characterization of which we address in this letter, is akin to the space that is achieved by subtracting channelized portions contributed due to concave regions from the watershed space (X). This non-network space is similar to non-channelized convex region within a catchment basin. We propose an alternative shape-dependent quantity like fractal dimension to characterize this non-network space. Towards this goal, we decompose the non-network space in two-dimensional discrete space into simple non-overlapping disks (NODs) of various sizes by employing mathematical morphological transformations and certain logical operations. Furthermore, we plot the number of NODs of less than threshold radius against the radius, and compute the shape-dependent fractal dimension of non-network space.