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.

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 { π₀( X \ K ) }, where π₀(Y) denotes the set of connected components of a space Y, and each inclusion map Y → Z induces a function π₀(Y) → π₀(Z). Then set of ends of X is defined to be the inverse limit of this inverse system.

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