Subsurfaces

A subsurface is a surface that lies inside another surface as a subset, equipped with the surface structure inherited from the ambient surface. If S is a two-dimensional manifold (possibly with boundary), a subset A⊆S is called a subsurface when A itself carries a manifold structure whose atlas consists of restrictions of charts from S. In practice, A inherits its topology and smooth structure from S via the subspace topology.

The boundary of a subsurface is determined by its intersection with the ambient surface. If S has boundary, a subsurface may acquire boundary components that lie along or inside S’s boundary. A subsurface is typically required to be a surface in its own right, which means every point of A has a neighborhood in A homeomorphic to an open disk (interior points) or to a half-disk (boundary points). Subsurfaces can be connected or disconnected, compact or noncompact, and may be with or without boundary depending on how they sit inside S.

Common ways to obtain subsurfaces include removing open disks from S, cutting along simple closed curves, or taking closures of regions bounded by these curves. Classic examples include a torus with a disk removed (a once-punctured torus) or a pair of pants (a sphere with three boundary components). In the study of surfaces, subsurfaces are central to decompositions like pants decompositions and to the analysis of mapping class groups and Teichmüller spaces.

In summary, a subsurface is an embedded surface within a larger surface, naturally inheriting its smooth and topological structure, with boundary behavior determined by its position inside the ambient surface.