Interiority, in relation to topology, is understood as an internalized centralization of space, a condition with an inherent spatial intimacy. Interiority is increasingly eroded in contemporary architectural practice; this enclosure has been methodically worn away. 1 The formal boundary of the interior, the interface of the inside to the outside, has in the past century theoretically and formally evolved from a defined threshold to a porous, ambiguous gradient. This chapter will focus on a specific aspect of this erosion, namely topological surfaces, and explore the characteristics of this gradient and speculate on strategies to re-establish interiority through topological geometries.