endsurface

Endsurface is not a standard term in mathematics or related disciplines, but it is sometimes used informally to refer to the end structure of a surface. In topology, the ends of a non-compact surface describe how the surface behaves at infinity and are treated as a key invariant alongside genus and orientability.

An end of a surface is an equivalence class of nested sequences of nonempty open connected subsets that “go off to infinity.” Intuitively, an end represents a distinct way to travel toward infinity on the surface without returning to a compact region. The collection of all ends of a surface S forms the end space E(S), which is endowed with a natural topology making E(S) a compact, totally disconnected space. The end space can be finite, countably infinite, or possess a Cantor-set structure, and its configuration is central to classifying infinite-type surfaces.

Non-compact surfaces are often classified by the pair (genus, end space). This means that two surfaces are considered the same type if they have the same genus distribution and the same end space, together with whether ends are planar or non-planar in certain contexts. The end space can be decomposed into regions corresponding to different ends, including isolated ends and accumulation points of ends.

Examples help illustrate the concept: the plane has one end; a finite cylinder has two ends; the plane minus a Cantor set has as end space a Cantor set; the Loch Ness monster surface has one end with infinite genus.

In practice, ends are used in the study of non-compact surfaces in topology and geometric group theory, and in the construction of end-compactifications, which formally attach ends as boundary-like points to obtain compact spaces.