en>Michael Hardy at 03:24, 8 February 2013

2013-02-08T03:24:38Z

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	~~I would like~~ to ~~introduce myself~~ to ~~you~~, ~~I am Andrew~~ and ~~my wife doesn~~'~~t like it at all. He~~ is an ~~order clerk and~~ [~~http:~~//~~alles-herunterladen~~.de/~~excellent-advice-~~for~~-picking-~~the~~-ideal-hobby/ real psychics~~] it'~~s something he really appreciate. For many years she~~'s ~~been residing in Kentucky but her husband wants them to transfer~~. ~~What me and my family love~~ is ~~to climb but I~~'~~m considering on beginning something new.~~<br><br>~~Look at my blog~~; ~~love psychic~~ (~~[http:~~//si.~~dgmensa~~.~~org~~/xe/~~index.php?document_srl~~=~~48014~~&~~mid~~=~~c0102 http:~~//si.~~dgmensa~~.~~org~~/xe/~~index~~.~~php?document_srl~~=~~48014&mid~~=~~c0102~~]) ~~best psychics~~ ([~~http~~:/~~/galab~~-~~work~~.cs.~~pusan~~.ac.kr/~~Sol09B~~/~~?document_srl~~=~~1489804 mouse click the up coming webpage~~])		In [[topology]], a branch of [[mathematics]], the '''ends''' of a [[topological space]] are, roughly speaking, the [[connected component (topology)\|connected components]] of the “ideal boundary” of the space. That is, each end represents a topologically distinct way to move to [[infinity]] within the space. Adding a point at each end yields a [[Compactification (mathematics)\|compactification]] of the original space, known as the '''end compactification'''.

			==Definition==

			Let ''X'' be a [[topological space]], and suppose that
			: ''K''<sub>1</sub> ⊂ ''K''<sub>2</sub> ⊂ ''K''<sub>3</sub> ⊂ · · ·
			is an ascending sequence of [[compact subset]]s of ''X'' whose [[interior (topology)\|interiors]] [[cover (topology)\|cover]] ''X''. Then ''X'' has one '''end''' for every sequence
			: ''U''<sub>1</sub> ⊃ ''U''<sub>2</sub> ⊃ ''U''<sub>3</sub> ⊃ · · ·,
			where each ''U''<sub>''n''</sub> is a [[connected component (topology)\|connected component]] of ''X'' \ ''K''<sub>''n''</sub>. The number of ends does not depend of the specific sequence {''K''<sub>''i''</sub>} of compact sets; in fact, there is a [[natural transformation\|natural]] [[bijection]] between the sets of ends associated with any two such sequences.

			Using this definition, a '''neighborhood''' of an end {''U''<sub>''i''</sub>} is an open set ''V'' such that ''V'' ⊃ ''U''<sub>''n''</sub> for some ''n''. Such neighborhoods represent the neighborhoods of the corresponding point at infinity in the '''end compactification''' (this “compactification” isn’t always compact, the topological space ''X'' has to be connected and locally connected)

			The definition of ends given above applies only to spaces ''X'' that possess an [[exhaustion by compact sets]] (that is, ''X'' must be [[hemicompact space\|hemicompact]]). However, it can be generalized as follows: let ''X'' be any topological space, and consider the [[direct system (mathematics)\|direct system]] {''K''<sub>α</sub>} of compact subsets of ''X'' and [[inclusion map]]s. There is a corresponding [[inverse system]] { ''π''<sub>0</sub>( ''X'' \ ''K''<sub>α</sub> ) }, where ''π''<sub>0</sub>(''Y'') denotes the set of connected components of a space ''Y'', and each inclusion map ''Y'' → ''Z'' induces a function ''π''<sub>0</sub>(''Y'') → ''π''<sub>0</sub>(''Z''). Then '''set of ends''' of ''X'' is defined to be the [[inverse limit]] of this inverse system. Under this definition, the set of ends is a [[functor]] from the [[category of topological spaces]] to the [[category of sets]]. The original definition above represents the special case where the direct system of compact subsets has a [[cofinal sequence]].

			==Examples==
			* The set of ends of any [[compact space]] is the [[empty set]].
			* The [[real line]] <math>\scriptstyle \mathbb{R} </math> has two ends. For example, if we let ''K''<sub>''n''</sub> be the [[closed interval]] [−''n'', ''n''], then the two ends are the sequences of open sets ''U''<sub>''n''</sub> = (''n'', ∞) and ''V''<sub>''n''</sub> = (−∞, −''n''). These ends are usually referred to as “infinity” and “minus infinity”, respectively.
			* If ''n'' > 1, then the Euclidean space <math>\scriptstyle \mathbb{R}^n </math> has only one end. This is because <math>\scriptstyle \mathbb{R}^n \setminus K</math> has only one unbounded component for any compact set ''K''.
			* More generally, if ''M'' is a compact [[manifold with boundary]], then the number of ends of the interior of ''M'' is equal to the number of connected components of the boundary of ''M''.
			* The union of ''n'' distinct [[ray (mathematics)\|rays]] emanating from the origin in <math>\scriptstyle \mathbb{R}^2 </math> has ''n'' ends.
			* The [[binary tree#Types of binary trees\|infinite complete binary tree]] has uncountably many ends, corresponding to the uncountably many different descending paths starting at the root. (This can be seen by letting ''K''<sub>''n''</sub> be the complete binary tree of depth ''n''.) These ends can be thought of as the “leaves” of the infinite tree. In the end compactification, the set of ends has the topology of a [[Cantor set]].

			==History==
			The notion of an end of a topological space was introduced by {{harvs\|first=Hans\|last=Freudenthal\|authorlink=Hans Freudenthal\|year=1931\|txt}}.

			==Ends of graphs and groups==
			{{main\|End (graph theory)}}
			In [[infinite graph\|infinite]] [[graph theory]], an end is defined slightly differently, as an equivalence class of semi-infinite paths in the graph, or as a [[Haven (graph theory)\|haven]], a function mapping finite sets of vertices to connected components of their complements. However, for locally finite graphs (graphs in which each vertex has finite [[degree (graph theory)\|degree]]), the ends defined in this way correspond one-for-one with the ends of topological spaces defined from the graph {{harv\|Diestel\|Kühn\|2003}}.

			The ends of a [[finitely generated group]] are defined to be the ends of the corresponding [[Cayley graph]]; this definition is insensitive to the choice of generating set. Every finitely-generated infinite group has either 1, 2, or infinitely many ends, and [[Stallings theorem about ends of groups]] provides a decomposition for groups with more than one end.

			==Ends of a CW complex==
			For a [[path connected]] [[CW-complex]], the ends can be characterized as [[homotopy classes]] of [[proper map]]s <math>\scriptstyle\mathbb{R}^+\to X</math>, called [[Line (mathematics)\|rays]] in ''X'': more precisely, if between the restriction -to the subset <math>\scriptstyle\mathbb{N}</math>- 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==
			*{{citation
			\| last1 = Diestel \| first1 = Reinhard
			\| last2 = Kühn \| first2 = Daniela
			\| doi = 10.1016/S0095-8956(02)00034-5
			\| issue = 1
			\| journal = [[Journal of Combinatorial Theory]] \| series = Series B
			\| mr = 1967888
			\| pages = 197–206
			\| title = Graph-theoretical versus topological ends of graphs
			\| volume = 87
			\| year = 2003}}.
			*{{Citation \| last1=Freudenthal \| first1=Hans \| author1-link=Hans Freudenthal \| title=Über die Enden topologischer Räume und Gruppen \| publisher=Springer Berlin / Heidelberg \| doi=10.1007/BF01174375 \| zbl=0002.05603 \| year=1931 \| journal=[[Mathematische Zeitschrift]] \| issn=0025-5874 \| volume=33 \| pages=692–713}}
			* 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.

			[[Category:General topology]]
			[[Category:Properties of topological spaces]]
			[[Category:Compactification]]

en>Hmains: copyedit, clarity edits, MOS implementation, and/or AWB general fixes using AWB

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en>Michael Hardy at 03:24, 8 February 2013

en>Hmains: copyedit, clarity edits, MOS implementation, and/or AWB general fixes using AWB