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The boundary of a set in topology. The boundary operator on a chain complex in homological algebra. The boundary operator of a differential graded algebra. The conjugate of the Dolbeault operator on complex differential forms. The boundary ∂(S) of a set of vertices S in a graph is the set of edges leaving S, which defines a cut.
A boundary point of a set is any element of that set's boundary. The boundary 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.
The meaning of binding operators is supplied by the semantics of the language and does not concern us here. Variable binding relates three things: a variable v, a location a for that variable in an expression and a non-leaf node n of the form Q(v, P). Note: we define a location in an expression as a leaf node in the syntax tree.
This concept has since been widely cited and the concept of a boundary object has been adopted in computer science (particularly computer supported cooperative work), information science, [4] and management, particularly when considering cross-disciplinary work and collaboration, [5] either within one organization or with the boundary object helping to focus the efforts of multiple organizations.
Conversely, a set which is not bounded is called unbounded. The word "bounded" makes no sense in a general topological space without a corresponding metric. Boundary is a distinct concept; for example, a circle (not to be confused with a disk) in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
According to this definition the boundary of a subset S is different from the boundary of the complement I – S which is a topological paradox. To define the boundary correctly it is necessary to introduce a topological space corresponding to the given digital image. Such space can be a two-dimensional abstract cell complex.
The covering number () is the minimal number of open balls of radius required to cover the fractal, or in other words, such that their union contains the fractal. We can also consider the intrinsic covering number N covering ′ ( ε ) {\textstyle N'_{\text{covering}}(\varepsilon )} , which is defined the same way but with the additional ...
A "k-dimensional hole" is a k-dimensional cycle that is not a boundary of a (k+1)-dimensional object. The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional simplicial complexes: b 0 is the number of connected components; b 1 is the number of one-dimensional or "circular" holes;