Ends of a space
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Let X be a non-compact topological space. Suppose that K is a non-empty compact subset of X, and 
a connected component of 
, and V ⊆ U ⊆ X an open set containing V. Then U is a neighborhood of an end of X.
An end of X is an equivalence class of sequences 
such that 
, where Ui is a neighborhood of an end.
Two such sequences (Ui),(Vj) are equivalent if for all i, there exists j such that 
, and for all j, there exists i such that 
. Given an end 
and a neighborhood of an end U, U is called a neighborhood of if there is a sequence (Ui) such that ![[(U_i)]=\mathcal{E}](../../../math/f/e/d/fedb080a780fac7cd9654ecb22606543.png)
and 
. The notion of an end of a topological space was introduced by Hans Freudenthal.
For example, 
has two ends, with ends given by ![\left[( (i, \infty) )_{i\in\mathbb{N}}\right], \left[((-\infty, -i))_{i\in\mathbb{N}}\right]](../../../math/6/7/9/679aec4a8a55bfa9102860de6ed8d96f.png)
.
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