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| {{about|[[topology]] and [[mathematics]]|interfaces in [[Topological space|topological spaces]]|manifold}}
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| [[Image:Runge theorem.svg|right|thumb|A set (in light blue) and its boundary (in dark blue).]]
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| In [[topology]] and [[mathematics]] in general, the '''boundary''' of a subset ''S'' of a [[topological space]] ''X'' is the set of points which can be approached both from ''S'' and from the outside of ''S''. More precisely, it is the set of points in the [[closure (topology)|closure]] of ''S'', not belonging to the [[interior (topology)|interior]] of ''S''. An element of the boundary of ''S'' is called a '''boundary point''' of ''S''. Notations used for boundary of a set ''S'' include bd(''S''), fr(''S''), and ∂''S''. Some authors (for example Willard, in ''General Topology'') use the term '''frontier''', instead of boundary in an attempt to avoid confusion with the concept of boundary used in [[algebraic topology]] and [[manifold theory]]. However, frontier sometimes refers to a different set, which is the set of boundary points which are not actually in the set; that is, <span style="text-decoration: overline">''S''</span> \ ''S''.
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| A [[Connected_space#Formal_definition|connected component]] of the boundary of ''S'' is called a '''boundary component''' of ''S''.
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| == Common definitions ==
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| There are several common (and equivalent) definitions to the boundary of a subset ''S'' of a topological space ''X'':
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| *the closure of ''S'' without the interior of ''S'': ∂''S'' = <span style="text-decoration: overline">''S''</span> \ ''S''<sup>o</sup>.
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| *the intersection of the closure of ''S'' with the closure of its [[complement (set theory)|complement]]: ∂''S'' = <span style="text-decoration: overline">''S''</span> ∩ <span style="text-decoration: overline">(''X'' \ ''S'')</span>.
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| *the set of points ''p'' of ''X'' such that every [[neighborhood (topology)|neighborhood]] of ''p'' contains at least one point of ''S'' and at least one point not of ''S''.
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| == Examples ==
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| [[Image:Mandelbrot_set_Components.jpg|right|thumb|Boundary of hyperbolic components of [[Mandelbrot set]]]]
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| Consider the real line '''R''' with the usual topology (i.e. the topology whose [[basis (topology)|basis sets]] are [[open interval]]s). One has
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| * ∂(0,5) = ∂[0,5) = ∂(0,5] = ∂[0,5] = {0,5}
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| * ∂∅ = ∅
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| * ∂'''Q''' = '''R'''
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| * ∂('''Q''' ∩ [0,1]) = [0,1]
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| These last two examples illustrate the fact that the boundary of a [[dense set]] with empty interior is its closure.
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| In the space of rational numbers with the usual topology (the [[subspace topology]] of '''R'''), the boundary of <math>(-\infty, a)</math>, where ''a'' is irrational, is empty.
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| The boundary of a set is a [[topology|topological]] notion and may change if one changes the topology. For example, given the usual topology on '''R'''<sup>2</sup>, the boundary of a closed disk Ω = {(''x'',''y'') | ''x''<sup>2</sup> + ''y''<sup>2</sup> ≤ 1} is the disk's surrounding circle: ∂Ω = {(''x'',''y'') | ''x''<sup>2</sup> + ''y''<sup>2</sup> = 1}. If the disk is viewed as a set in '''R'''<sup>3</sup> with its own usual topology, i.e. Ω = {(''x'',''y'',0) | ''x''<sup>2</sup> + ''y''<sup>2</sup> ≤ 1}, then the boundary of the disk is the disk itself: ∂Ω = Ω. If the disk is viewed as its own topological space (with the subspace topology of '''R'''<sup>2</sup>), then the boundary of the disk is empty.
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| ==Properties==
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| * The boundary of a set is [[closed set|closed]].
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| * The boundary of a set is the boundary of the complement of the set: ∂''S'' = ∂(''S<sup>C</sup>'').
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| Hence:
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| * ''p'' is a boundary point of a set if and only if every neighborhood of ''p'' contains at least one point in the set and at least one point not in the set.
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| * A set is closed if and only if it contains its boundary, and [[open set|open]] if and only if it is disjoint from its boundary.
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| * The closure of a set equals the union of the set with its boundary. <span style="text-decoration:overline">''S''</span> = ''S'' ∪ ∂''S''.
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| * The boundary of a set is empty if and only if the set is both closed and open (that is, a [[clopen set]]).
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| * In '''R'''<sup>n</sup>, every closed set is the boundary of some open set.
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| ::::[[Image:AccumulationAndBoundaryPointsOfS.PNG]]
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| :''[[Concept]]ual [[Venn diagram]] showing the relationships among different points of a subset S of '''R'''<sup>n</sup>. A = set of [[limit point]]s of S, B = set of '''boundary points''' of S, area shaded green = set of [[interior points]] of S, area shaded yellow = set of [[isolated point]]s of S, areas shaded black = empty sets. Every point of S is either an interior point or a boundary point. Also, every point of S is either an accumulation point or an isolated point. Likewise, every boundary point of S is either an accumulation point or an isolated point. Isolated points are always boundary points.'' | |
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| == Boundary of a boundary == | |
| For any set ''S'', ∂''S'' ⊇ ∂∂''S'', with equality holding if and only if the boundary of ''S'' has no interior points, which will be the case for example if ''S'' is either closed or open. Since the boundary of a set is closed, ∂∂''S'' = ∂∂∂''S'' for any set ''S''. The boundary operator thus satisfies a weakened kind of [[idempotence]].
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| In discussing boundaries of [[manifold]]s or [[simplex]]es and their [[simplicial complex]]es, one often meets the assertion that the boundary of the boundary is always empty. Indeed, the construction of the [[singular homology]] rests critically on this fact. The explanation for the apparent incongruity is that the topological boundary (the subject of this article) is a slightly different concept than the boundary of a manifold or of a simplicial complex. For example, the topological boundary of a closed disk viewed as a topological space is empty, while its boundary in the sense of manifolds is the circle surrounding the disk.
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| ==See also==
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| *See the discussion of boundary in [[topological manifold]] for more details.
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| *[[Lebesgue's density theorem]], for measure-theoretic characterization and properties of boundary
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| *[[bounding point]]
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| == References ==
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| *{{cite book | author = J. R. Munkres | title = Topology | publisher = Prentice-Hall | year = 2000 | isbn=0-13-181629-2 }}
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| *{{cite book | author = S. Willard | title = General Topology | publisher = Addison-Wesley | year = 1970 | isbn=0-201-08707-3 }}
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| *{{cite book | author = L. van den Dries | title = Tame Topology | year = 1998 | isbn=978-0521598385 }}
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| [[Category:General topology]]
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