All points not part of the interior of a subset of a topological space
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include formula_1 and formula_2.
Some authors (for example Willard, in "General Topology") use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, "Metric Spaces" by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.
Definitions.
There are several equivalent definitions for the boundary of a subset formula_3 of a topological space formula_4 which will be denoted by formula_5 formula_6 or simply formula_2 if formula_8 is understood:
A boundary point of a set is any element of that set's boundary. The boundary formula_9 defined above is sometimes called the set's topological boundary to distinguish it from other similarly named notions such as the boundary of a manifold with boundary or the boundary of a manifold with corners, to name just a few examples.
A connected component of the boundary of S is called a boundary component of S.
Properties.
The closure of a set formula_10 equals the union of the set with its boundary: 
formula_11 
where formula_12 denotes the closure of formula_10 in formula_14 
A set is closed if and only if it contains its boundary, and open if and only if it is disjoint from its boundary. The boundary of a set is closed; this follows from the formula formula_15 which expresses formula_9 as the intersection of two closed subsets of formula_14 
("Trichotomy") Given any subset formula_18 each point of formula_8 lies in exactly one of the three sets formula_20 and formula_21 Said differently, formula_22 and these three sets are pairwise disjoint. Consequently, if these set are not empty then they form a partition of formula_14 
A point formula_24 is a boundary point of a set if and only if every neighborhood of formula_25 contains at least one point in the set and at least one point not in the set. 
The boundary of the interior of a set as well as the boundary of the closure of a set are both contained in the boundary of the set. 
"Conceptual Venn diagram showing the relationships among different points of a subset formula_10 of formula_27 formula_28 = set of accumulation points of formula_10 (also called limit points), formula_30 set of boundary points of formula_31 area shaded green = set of interior points of formula_31 area shaded yellow = set of isolated points of formula_31 areas shaded black = empty sets. Every point of formula_10 is either an interior point or a boundary point. Also, every point of formula_10 is either an accumulation point or an isolated point. Likewise, every boundary point of formula_10 is either an accumulation point or an isolated point. Isolated points are always boundary points."
Examples.
Characterizations and general examples.
A set and its complement have the same boundary:
formula_37
A set formula_38 is a dense open subset of formula_8 if and only if formula_40 
The interior of the boundary of a closed set is empty. 
Consequently, the interior of the boundary of the closure of a set is empty. 
The interior of the boundary of an open set is also empty. 
Consequently, the interior of the boundary of the interior of a set is empty. 
In particular, if formula_3 is a closed or open subset of formula_8 then there does not exist any nonempty subset formula_43 such that formula_38 is open in formula_14 
This fact is important for the definition and use of nowhere dense subsets, meager subsets, and Baire spaces. 
A set is the boundary of some open set if and only if it is closed and nowhere dense. 
The boundary of a set is empty if and only if the set is both closed and open (that is, a clopen set).
Concrete examples.
Consider the real line formula_46 with the usual topology (that is, the topology whose basis sets are open intervals) and formula_47 the subset of rational numbers (whose topological interior in formula_46 is empty). Then
These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure. They also show that it is possible for the boundary formula_2 of a subset formula_10 to contain a non-empty open subset of formula_55; that is, for the interior of formula_2 in formula_8 to be non-empty. However, a closed subset's boundary always has an empty interior. 
In the space of rational numbers with the usual topology (the subspace topology of formula_46), the boundary of formula_59 where formula_60 is irrational, is empty.
The boundary of a set is a topological notion and may change if one changes the topology. For example, given the usual topology on formula_61 the boundary of a closed disk formula_62 is the disk's surrounding circle: formula_63 If the disk is viewed as a set in formula_64 with its own usual topology, that is, formula_65 then the boundary of the disk is the disk itself: formula_66 If the disk is viewed as its own topological space (with the subspace topology of formula_67), then the boundary of the disk is empty. 
Boundary of an open ball vs. its surrounding sphere.
This example demonstrates that the topological boundary of an open ball of radius formula_68 is not necessarily equal to the corresponding sphere of radius formula_69 (centered at the same point); it also shows that the closure of an open ball of radius formula_68 is not necessarily equal to the closed ball of radius formula_69 (again centered at the same point). 
Denote the usual Euclidean metric on formula_67 by
formula_73
which induces on formula_67 the usual Euclidean topology. 
Let formula_75 denote the union of the formula_76-axis formula_77 with the unit circle formula_78 centered at the origin formula_79; that is, formula_80 which is a topological subspace of formula_67 whose topology is equal to that induced by the (restriction of) the metric formula_82 
In particular, the sets formula_83 and formula_84 are all closed subsets of formula_67 and thus also closed subsets of its subspace formula_14 
Henceforth, unless it clearly indicated otherwise, every open ball, closed ball, and sphere should be assumed to be centered at the origin formula_87 and moreover, only the metric space formula_88 will be considered (and not its superspace formula_89); this being a path-connected and locally path-connected complete metric space. 
Denote the open ball of radius formula_68 in formula_88 by 
formula_92
so that when formula_93 then 
formula_94
is the open sub-interval of the formula_76-axis strictly between formula_96 and formula_97 
The unit sphere in formula_88 ("unit" meaning that its radius is formula_93) is 
formula_100
while the closed unit ball in formula_88 is the union of the open unit ball and the unit sphere centered at this same point: 
formula_102
However, the topological boundary formula_103 and topological closure formula_104 in formula_8 of the open unit ball formula_106 are:
formula_107
In particular, the open unit ball's topological boundary formula_108 is a proper subset of the unit sphere formula_109 in formula_110 
And the open unit ball's topological closure formula_111 is a proper subset of the closed unit ball formula_112 in formula_110 
The point formula_114 for instance, cannot belong to formula_104 because there does not exist a sequence in formula_116 that converges to it; the same reasoning generalizes to also explain why no point in formula_8 outside of the closed sub-interval formula_84 belongs to formula_119 Because the topological boundary of the set formula_106 is always a subset of formula_106's closure, it follows that formula_103 must also be a subset of formula_123 
In any metric space formula_124 the topological boundary in formula_125 of an open ball of radius formula_68 centered at a point formula_127 is always a subset of the sphere of radius formula_69 centered at that same point formula_129; that is, 
formula_130
always holds. 
Moreover, the unit sphere in formula_88 contains formula_132 which is an open subset of formula_14 This shows, in particular, that the unit sphere formula_134 in formula_88 contains a non-empty open subset of formula_14
Boundary of a boundary.
For any set formula_137 where formula_138 denotes the superset with equality holding if and only if the boundary of formula_10 has no interior points, which will be the case for example if formula_10 is either closed or open. Since the boundary of a set is closed, formula_141 for any set formula_142 The boundary operator thus satisfies a weakened kind of idempotence.
In discussing boundaries of manifolds or simplexes and their simplicial complexes, 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 from the boundary of a manifold or of a simplicial complex. For example, the boundary of an open disk viewed as a manifold is empty, as is its topological boundary viewed as a subset of itself, while its topological boundary viewed as a subset of the real plane is the circle surrounding the disk. Conversely, the boundary of a closed disk viewed as a manifold is the bounding circle, as is its topological boundary viewed as a subset of the real plane, while its topological boundary viewed as a subset of itself is empty. In particular, the topological boundary depends on the ambient space, while the boundary of a manifold is invariant.
Notes.
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Citations.
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