In topology, a branch of mathematics, the ends of a topological space are, roughly speaking, the connected components of the "ideal boundary" of the space.
That is, each end represents a topologically distinct way to move to infinity within the space.
Adding a point at each end yields a compactification of the original space, known as the end compactification.
The notion of an end of a topological space was introduced by Hans Freudenthal (1931).
be a topological space, and suppose that is an ascending sequence of compact subsets of
has one end for every sequence where each
The number of ends does not depend on the specific sequence
of compact sets; there is a natural bijection between the sets of ends associated with any two such sequences.
Using this definition, a neighborhood of an end
Such neighborhoods represent the neighborhoods of the corresponding point at infinity in the end compactification (this "compactification" is not always compact; the topological space X has to be connected and locally connected).
The definition of ends given above applies only to spaces
that possess an exhaustion by compact sets (that is,
be any topological space, and consider the direct system
There is a corresponding inverse system
denotes the set of connected components of a space
Then set of ends of
is defined to be the inverse limit of this inverse system.
Under this definition, the set of ends is a functor from the category of topological spaces, where morphisms are only proper continuous maps, to the category of sets.
in the family is a connected component of
and they are compatible with maps induced by inclusions) then
ranges over compact subsets of Y and
The original definition above represents the special case where the direct system of compact subsets has a cofinal sequence.
In infinite graph theory, an end is defined slightly differently, as an equivalence class of semi-infinite paths in the graph, or as a haven, a function mapping finite sets of vertices to connected components of their complements.
However, for locally finite graphs (graphs in which each vertex has finite degree), the ends defined in this way correspond one-for-one with the ends of topological spaces defined from the graph (Diestel & Kühn 2003).
The ends of a finitely generated group are defined to be the ends of the corresponding Cayley graph; this definition is insensitive to the choice of generating set.
Every finitely-generated infinite group has either 1, 2, or infinitely many ends, and Stallings theorem about ends of groups provides a decomposition for groups with more than one end.
For a path connected CW-complex, the ends can be characterized as homotopy classes of proper maps
, called rays in X: more precisely, if between the restriction —to the subset
— of any two of these maps exists a proper homotopy we say that they are equivalent and they define an equivalence class of proper rays.
This set is called an end of X.