End (topology)

In topology, a branch of mathematics, an end of a topological space is a point in a certain kind of compactification of the space. But also as a way to approach infinity within the space.

The definition

Let "X" be a non-compact topological space. Suppose that "K" is a non-empty compact subset of "X", and $$scriptstyle V subseteq Xackslash K a connected component of $$scriptstyle Xackslash K, 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 $$scriptstyle X supset U_1 supset U_2 supset cdots such that $$scriptstylecap overline{U}_i = varnothing, where $$U_i is a neighborhood of an end.

Two such sequences $$scriptstyle {U_i}, {V_j} are equivalent if forall "i", there exists "j" such that $$scriptstyle U_i supset V_j,and for all "j", there exists "i" such that $$scriptstyle V_j supset U_i .Given an end $$scriptstyle mathcal{E} and a neighborhood of an end $$U,$$U is called a neighborhood of $$scriptstylemathcal{E} if there isa sequence $$scriptstyle {U_i} such that $$scriptstyle [{U_i}] =mathcal{E} and $$scriptstyle U_1 subset U.

History

The notion of an end of a topological space was introduced by Hans Freudenthal.

Examples

For example, $$scriptstyle mathbb{R} has two ends, with endsgiven by $$scriptstyle left [( (n, infty) )_{ninmathbb{N ight] , left [((-infty, -n))_{ninmathbb{N ight] .

But in $$scriptstyle mathbb{R}^n with "n" greater than one we are going to have only one end.

Further

Ends can be characterized in a number of ways using algebraic functors.

* For example, the set of compact subsets of $$X is partially ordered by inclusion. Taking complements defines a partial order on the set of complements $$scriptstyle X ackslash K where $$K ranges over all compact sets. An inclusion $$scriptstyle K o L of compact sets induces a map, using the $$pi_0 functor, from $$scriptstyle Xackslash L o Xackslash K. The inverse limit ::$$scriptstylelim_{leftarrow} pi_0 (Xackslash K) :over all compact subsets $$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 $$scriptstylemathbb{R}^+ o X, called rays in "X": more precisely, if between the restriction -to the subset $$scriptstylemathbb{N}- 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".

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.