en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

2014-01-31T03:32:13Z

References: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 05:32, 31 January 2014
Line 1:		Line 1:
	~~Hello~~ and ~~welcome~~. ~~My name~~ is ~~Figures Wunder~~. ~~My day job~~ is a ~~librarian~~. ~~Doing ceramics~~ is ~~what adore doing~~. ~~Years ago we moved~~ to ~~North Dakota~~ and I ~~adore each working day residing right here~~.<br><br>~~Also visit my web blog~~ ... [~~http~~://~~url~~.~~Skyost~~.eu/~~fooddeliveryservices36486 url~~.~~Skyost~~.eu]		[[Image:Path.svg\|thumb\|The points traced by a path from ''A'' to ''B'' in '''R'''². However, different paths can trace the same set of points.]]

			In [[mathematics]], a '''path''' in a [[topological space]] ''X'' is a [[continuous (topology)\|continuous map]] ''f'' from the [[unit interval]] ''I'' = [0,1] to ''X''
			:''f'' : ''I'' → ''X''.
			The ''initial point'' of the path is ''f''(0) and the ''terminal point'' is ''f''(1). One often speaks of a "path from ''x'' to ''y''" where ''x'' and ''y'' are the initial and terminal points of the path. Note that a path is not just a subset of ''X'' which "looks like" a [[curve]], it also includes a [[coordinate system\|parameterization]]. For example, the maps ''f''(''x'') = ''x'' and ''g''(''x'') = ''x''<sup>2</sup> represent two different paths from 0 to 1 on the real line.

			A '''[[Loop (topology)\|loop]]''' in a space ''X'' based at ''x'' ∈ ''X'' is a path from ''x'' to ''x''. A loop may be equally well regarded as a map ''f'' : ''I'' → ''X'' with ''f''(0) = ''f''(1) or as a continuous map from the [[unit circle]] ''S''<sup>1</sup> to ''X''
			:''f'' : ''S''<sup>1</sup> → ''X''.
			This is because ''S''<sup>1</sup> may be regarded as a [[quotient space\|quotient]] of ''I'' under the identification 0 ∼ 1. The set of all loops in ''X'' forms a space called the [[loop space]] of ''X''.

			A topological space for which there exists a path connecting any two points is said to be [[path-connected space\|path-connected]]. Any space may be broken up into a set of [[path-connected component]]s. The set of path-connected components of a space ''X'' is often denoted π<sub>0</sub>(''X'');.

			One can also define paths and loops in [[pointed space]]s, which are important in [[homotopy theory]]. If ''X'' is a topological space with basepoint ''x''<sub>0</sub>, then a path in ''X'' is one whose initial point is ''x''<sub>0</sub>. Likewise, a loop in ''X'' is one that is based at ''x''<sub>0</sub>.

			==Homotopy of paths==
			{{Main\|Homotopy}}
			[[Image:Homotopy between two paths.svg\|thumb\|right\|A homotopy between two paths.]]
			Paths and loops are central subjects of study in the branch of [[algebraic topology]] called [[homotopy theory]]. A [[homotopy]] of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed.

			Specifically, a homotopy of paths, or '''path-homotopy''', in ''X'' is a family of paths ''f''<sub>''t''</sub> : ''I'' → ''X'' indexed by ''I'' such that
			* ''f''<sub>''t''</sub>(0) = ''x''<sub>0</sub> and ''f''<sub>''t''</sub>(1) = ''x''<sub>1</sub> are fixed.
			* the map ''F'' : ''I'' × ''I'' → ''X'' given by ''F''(''s'', ''t'') = ''f''<sub>''t''</sub>(''s'') is continuous.
			The paths ''f''<sub>0</sub> and ''f''<sub>1</sub> connected by a homotopy are said to '''homotopic''' (or more precisely '''path-homotopic''', to distinguish between the relation defined on all continuous functions between fixed spaces). One can likewise define a homotopy of loops keeping the base point fixed.

			The relation of being homotopic is an [[equivalence relation]] on paths in a topological space. The [[equivalence class]] of a path ''f'' under this relation is called the '''homotopy class''' of ''f'', often denoted [''f''].

			==Path composition==
			One can compose paths in a topological space in an obvious manner. Suppose ''f'' is a path from ''x'' to ''y'' and ''g'' is a path from ''y'' to ''z''. The path ''fg'' is defined as the path obtained by first traversing ''f'' and then traversing ''g'':
			:<math>fg(s) = \begin{cases}f(2s) & 0\leq s \leq \frac{1}{2} \\ g(2s-1) & \frac{1}{2} \leq s \leq 1.\end{cases}</math>
			Clearly path composition is only defined when the terminal point of ''f'' coincides with the initial point of ''g''. If one considers all loops based at a point ''x''<sub>0</sub>, then path composition is a [[binary operation]].

			Path composition, whenever defined, is not [[associative]] due to the difference in parametrization. However it ''is'' associative up to path-homotopy. That is, [(''fg'')''h''] = [''f''(''gh'')]. Path composition defines a [[group (mathematics)\|group structure]] on the set of homotopy classes of loops based at a point ''x''<sub>0</sub> in ''X''. The resultant group is called the [[fundamental group]] of ''X'' based at ''x''<sub>0</sub>, usually denoted π<sub>1</sub>(''X'',''x''<sub>0</sub>).

			In situations calling for associativity of path composition "on the nose," a path in ''X'' may instead be defined as a continuous map from an interval [0,''a''] to X for any real ''a'' ≥ 0. A path ''f'' of this kind has a length \|''f''\| defined as ''a''. Path composition is then defined as before with the following modification:
			:<math>fg(s) = \begin{cases}f(s) & 0\leq s \leq \|f\| \\ g(s-\|f\|) & \|f\| \leq s \leq \|f\|+\|g\|\end{cases}</math>
			Whereas with the previous definition, ''f'', ''g'', and ''fg'' all have length 1 (the length of the domain of the map), this definition makes \|''fg''\| = \|''f''\| + \|''g''\|. What made associativity fail for the previous definition is that although (''fg'')''h'' and ''f''(''gh'') have the same length, namely 1, the midpoint of (''fg'')''h'' occurred between ''g'' and ''h'', whereas the midpoint of ''f''(''gh'') occurred between ''f'' and ''g''. With this modified definition (''fg'')''h'' and ''f''(''gh'') have the same length, namely \|''f''\|+\|''g''\|+\|''h''\|, and the same midpoint, found at (\|''f''\|+\|''g''\|+\|''h''\|)/2 in both (''fg'')''h'' and ''f''(''gh''); more generally they have the same parametrization throughout.

			==Fundamental groupoid==
			There is a [[category theory\|categorical]] picture of paths which is sometimes useful. Any topological space ''X'' gives rise to a [[category (mathematics)\|category]] where the objects are the points of ''X'' and the [[morphism]]s are the homotopy classes of paths. Since any morphism in this category is an [[isomorphism]] this category is a [[groupoid]], called the [[fundamental groupoid]] of ''X''. Loops in this category are the [[endomorphism]]s (all of which are actually [[automorphism]]s). The [[automorphism group]] of a point ''x''<sub>0</sub> in ''X'' is just the fundamental group based at ''X''. More generally, one can define the fundamental groupoid on any subset ''A'' of ''X'', using homotopy classes of paths joining points of ''A''. This is convenient for the [[Van Kampen's Theorem]].

			==References==

			* Ronald Brown, Topology and groupoids, Booksurge PLC, (2006).

			* Peter May, A concise course in algebraic topology, University of Chicago Press, (1999).

			* James Raymond Munkres, Topology 2ed, Prentice Hall, (2000).

			{{DEFAULTSORT:Path (Topology)}}
			[[Category:Topology]]
			[[Category:Homotopy theory]]

en>Edward: link halting problem using Find link

2012-07-27T18:43:55Z

link halting problem using Find link

New page

Hello and welcome. My name is Figures Wunder. My day job is a librarian. Doing ceramics is what adore doing. Years ago we moved to North Dakota and I adore each working day residing right here.<br><br>Also visit my web blog ... [http://url.Skyost.eu/fooddeliveryservices36486 url.Skyost.eu]

Turing reduction - Revision history

en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

en>Edward: link halting problem using Find link