In topology, 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 formally, it 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. 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. A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

Touch my juicy boundary its firm.

Common definitions

There are several common (and equivalent) definitions to the boundary of S:

• the closure of S without the interior of S: .
• the intersection of the closure of S with the closure of its complement:
• the set of all boundary points of S where a point p in X is a boundary point of S if every neighborhood of p contains at least one point of S and at least one point not in S.

In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... This is a glossary of some terms used in the branch of mathematics known as topology. ...

Examples

Consider the real line R with the usual topology (i.e. the topology whose basis sets are open intervals). One has In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...

These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure. In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of...

The boundary of a set is a topological notion and may change if one changes the topology. For example, given the usual topology on R², the boundary of a closed disk Ω={(x, y): x²+y² ≤ 1} is the disk's surrounding circle: ∂Ω = {(x, y) | x²+y² = 1}. If the disk is viewed as a set in R³ with its own usual topology, i.e. Ω={(x, y, 0): x²+y² ≤ 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 induced topology), then the boundary of the disk is empty. A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology). ...

Properties

• The boundary of a set is closed.
• The boundary of a set is the boundary of the complement of the set: .

Hence: In topology and related branches of mathematics, a closed set is a set whose complement is open. ...

• 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.
• 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 closure of a set equals the union of the set with its boundary. .
• The boundary of a set is empty if and only if the set is both closed and open (that is, a clopen set).
• In Rⁿ, every closed set is the boundary of some set.

Conceptual Venn diagram showing the relationships among different points of set S. A = set of accumulation points of S, B = set of boundary points of S, area shaded green = set of interior points of S, area shaded yellow = set of isolated points 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.

In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology, a clopen set (or closed-open set, a portmanteau word) in a topological space is a set which is both open and closed. ... Image File history File links AccumulationAndBoundaryPointsOfS.PNG‎ I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... A concept is an abstract idea or a mental symbol, typically associated with a corresponding representation in language or symbology, that denotes all of the objects in a given category or class of entities, interactions, phenomena, or relationships between them. ... A Venn diagram of sets A, B, and C Venn diagrams are illustrations used in the branch of mathematics known as set theory. ... In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ... In topology, a point x of a set S is called an isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in an Euclidean space (or in a metric space), x is an isolated point of S, if one can find an...

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. This is always true if S is either closed or open. Since the boundary of any set is closed, ∂∂S=∂∂∂S for any set S. The boundary operator thus satisfies a weakened kind of idempotence. In particular, the boundary of the boundary of a set will usually be nonempty. There are two main definitions of idempotence (IPA , like eye-dem-potent-s) in mathematics. ...

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 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. See the discussion of boundary in topological manifold for more details. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In geometry, a simplex (plural: simplices) or n-simplex is an n-dimensional analogue of a triangle. ... In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ... In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...

References

• J. R. Munkres (2000). Topology. Prentice-Hall. ISBN 0-13-181629-2. 
• S. Willard (1970). General Topology. Addison-Wesley. ISBN 0-201-08707-3.