- 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 throughhomotopy classes ofproper map s 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.
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