- For a different notion of boundary related to manifolds, see that article. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be
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. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. More formally, it is the set of points in the closure of S, not belonging to the interior of S. In Mathematics, the closure of a set S consists of all points which are intuitively "close to S " In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge 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 connected component of the boundary of S is called a boundary component of S. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of
Contents |
Common definitions
There are several common (and equivalent) definitions to the boundary of a subset S of a topological space X:
- 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 points p of X such that every neighborhood of p contains at least one point of S and at least one point not of S. In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space.
Examples
Consider the real line R with the usual topology (i. e. the topology whose basis sets are open intervals). In Mathematics, a base (or basis) B for a Topological space X with topology T is a collection of Open sets In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set One has
![\partial (0,5) = \partial [0,5) = \partial (0,5] = \partial [0, 5] = \{0,5 \}\,](../../../../math/6/5/1/651bf94fcd4a09e983e9f3bda9b250da.png)
![\partial \big(\mathbb{Q}\cap\left[0,1\right]\big) = \left[0,1\right]](../../../../math/8/8/1/8810962287ecdb0c8b638f51aaa1c551.png)
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
In the space of rational numbers with the usual topology (the subspace topology of R), the boundary of the set of numbers of which the square is less than 2 is empty. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is
The boundary of a set is a topological notion and may change if one changes the topology. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of 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. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is
Properties
- The boundary of a set is closed. In Topology and related branches of Mathematics, a closed set is a set whose complement is open.
- The boundary of a set is the boundary of the complement of the set:
.
Hence:
- 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. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in
- 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 Topology, a clopen set (or closed-open set, a Portmanteau word in a Topological space is a set which is both open and closed
- In Rn, every closed set is the boundary of some set.
-
-
- Conceptual Venn diagram showing the relationships among different points of a subset S of Rn. The term "concept" is traced back to 1554–60 ( l conceptum - something conceived but what is today termed "the classical theory of concepts" is the theory of Aristotle Venn diagrams or set diagrams are Diagrams that show all hypothetically possible Logical relations between a finite collection of sets (groups 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. In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated" In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S " In Topology, a branch of Mathematics, a point x of a set S is called an isolated point,if there exists a neighborhood of 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.
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. Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation In particular, the boundary of the boundary of a set will usually be nonempty.
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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Geometry, a simplex (plural simplexes or 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 constructed by "gluing together" points Line segments Indeed, the construction of the singular homology rests critically on this fact. In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space 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 also
- See the discussion of boundary in topological manifold for more details. In Mathematics, a topological manifold is a Hausdorff Topological space which looks locally like Euclidean space in a sense defined below
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.
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic