This paper describes physically and mathematically how the plasma-vacuum boundary of a tokamak plasma equilibrium can be perturbed to form a separatrix with an X-point, while having an otherwise negligible affect on the plasma equilibrium. A deliberate consequence of the technique is that the radial and poloidal extent of the perturbed region may be arbitrarily localized. This has useful theoretical and physical consequences, namely 1 it is possible to take any plasma equilibrium and modify the outermost flux surface to form a separatrix with one or more additional x-points in a rigorous way, 2 subsequent studies will be able to separate the effects of shaping from those due to topological changes associated with a separatrix, for example, a circular cross-section plasma may be modified to form a separatrix that is circular everywhere except for an arbitrarily localized region that is perturbed to form an x-point, 3 because the perturbation is arbitrarily localized, there is the possibility for modifying the stability of the edge, without affecting the bulk plasma properties or stability. Therefore the ideas presented here provide analytical and conceptual tools to study how a separatrix can affect plasma stability, and a potential experimental technique to study the stability of the plasma edge. The paper also investigates how the magnetic shear and the Mercier coefficient behave as a separatrix is approached, showing that for a nonzero toroidal current at the x-point, the Mercier coefficient always asymptotes to zero.