The level set and density methods for topology optimization are often perceived as two very different approaches. This has to some extent led to two competing research directions working in parallel with only little overlap and knowledge exchange. In this paper, we conjecture that this is a misconception and that the overlap and similarities are far greater than the differences. To verify this claim, we employ, without significant modifications, many of the base ingredients from the density method to construct a crisp interface level set optimization approach using a simple cut element method. That is, we use the same design field representation, the same projection filters, the same optimizer, and the same so-called robust approach as used in density-based optimization for length scale control. The only noticeable difference lies in the finite element and sensitivity analysis, here based on a cut element method, which provides an accurate tool to model arbitrary, crisp interfaces on a structured mesh based on the thresholding of a level set-or density-field. The presented work includes a heuristic hole generation scheme and we demonstrate the design approach on several numerical examples covering compliance minimization and a compliant force inverter. Finally, we provide our MATLAB code, downloadable from www.topopt.dtu.dk, to facilitate further extension of the proposed method to, e.g., multiphysics problems.