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 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 $scriptstyle\; X\; supset\; U\_1\; supset\; U\_2\; supset\; cdots$ 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 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 . 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 $scriptstylemathbb\{R\}^+\; o\; X$, 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.