End (topology)

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 .
Definition.
Let "X" be a topological space, and suppose that
is an ascending sequence of compact subsets of "X" whose interiors cover "X". Then "X" has one end for every sequence
where each "U""n" is a connected component of "X" \ "K""n". The number of ends does not depend on the specific sequence {"K""i"} 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 {"U""i"} is an open set "V" such that "V" ⊃ "U""n" for some "n". 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 "X" that possess an exhaustion by compact sets (that is, "X" must be hemicompact). However, it can be generalized as follows: let "X" be any topological space, and consider the direct system {"K"} of compact subsets of "X" and inclusion maps. There is a corresponding inverse system { "π"0( "X" \ "K" ) }, where "π"0("Y") denotes the set of connected components of a space "Y", and each inclusion map "Y" → "Z" induces a function "π"0("Y") → "π"0("Z"). Then set of ends of "X" 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. Explicitly, if φ : X → Y is a proper map and "x"=("x""K")K is an end of "X" (i.e. each element "x""K" in the family is a connected component of "X" ∖ "K" and they are compatible with maps induced by inclusions) then φ(x) is the family formula_3 where formula_4 ranges over compact subsets of "Y" and φ* is the map induced by φ from formula_5 to formula_6. Properness of φ is used to ensure that each φ-1("K") is compact in "X".
The original definition above represents the special case where the direct system of compact subsets has a cofinal sequence.
Ends of graphs and groups.
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 .
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.
Ends of a CW complex.
For a path connected CW-complex, the ends can be characterized as homotopy classes of proper maps formula_11, called rays in "X": more precisely, if between the restriction -to the subset formula_12- 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".