End (topology)
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In topology, a branch of mathematics, an end of a topological space is a point in a certain kind of compactification of the space. It can be viewed as a way to approach infinity within the space.
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[edit] The definition
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 U an open set containing V (V ⊆ U ⊆ X). 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 
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 
such that ![\scriptstyle[\{U_i\}]=\mathcal{E}](http://upload.wikimedia.org/math/e/6/b/e6b3fca5d6b3aeb50c62da5a0eba54aa.png)
and 
.
Another definition, which applies to groups, is as follows. Let us call a set of elements almost invariant if applying any other element of the group results in a new set which differs from the original set only by a finite number of elements. For instance, in the group of integers, the set of positive integers is almost invariant because adding any integer to them results in a set which includes only a finite number of negative integers or excludes only a finite number of positive integers. If the group can be decomposed into two infinite sets each of which is almost invariant (such as the positive integers and the non-positive integers), then it has at least two ends. It has exactly two ends if any other such decomposition consists of two sets which are almost equal to the two sets of the original decomposition (meaning that they differ by a finite number of elements).[1]
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This section requires expansion with: an explanation of how the integers have two ends according to the definition given higher up. |
[edit] History
The notion of an end of a topological space was introduced by Hans Freudenthal.
[edit] Examples
For example, 
has two ends, with ends given by ![\scriptstyle \left[( (n, \infty) )_{n\in\mathbb{N}}\right], \left[((-\infty, -n))_{n\in\mathbb{N}}\right]](http://upload.wikimedia.org/math/a/7/8/a78c260723cb92bc427987cac9777cc6.png)
.
But in 
with n greater than one we are going to have only one end.
[edit] Further
Ends can be characterized in a number of ways using algebraic functors.
- For example, the set of compact subsets of X is partially ordered by inclusion. Taking complements defines a partial order on the set of complements where K ranges over all compact sets. An inclusion
of compact sets induces a map, using the π0 functor, from 
. The inverse limit
-
- over all compact subsets K defines the set of ends as a topological space.
- Another, for a path connected CW-complex space, is through 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.
[edit] Notes
- ^ "Ends and Free Products of Groups" by Daniel E. Cohen, Mathematische Zeitschrift 114, 9-18 (1970).
[edit] References
- Ross Geoghegan, Topological methods in group theory, GTM-243 (2008), Springer ISBN 978-0-387-74611-1.
- Peter Scott, Terry Wall, Topological methods in group theory, London Math. Soc. Lecture Note Ser., 36, Cambridge Univ. Press (1979) 137-203.
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