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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.
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.
1. Boundary of a topological subspace: If S is a subspace of a topological space, then its boundary, denoted , is the set difference between the closure and the interior of S. 2. Partial derivative: see ∂ / ∂ . ∫ 1. Without a subscript, denotes an antiderivative.
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. A bounded set is not necessarily a closed set and vice versa.
Commonly considered types of domains are domains with continuous boundary, Lipschitz boundary, C 1 boundary, and so forth. A bounded domain is a domain that is bounded, i.e., contained in some ball. Bounded region is defined similarly.
Perimeter is the distance around a two dimensional shape, a measurement of the distance around something; the length of the boundary. A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
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Boundary turbulence, a break in expectation of how private information is held within a group of knowledge owners; Boundary Ale, a beer made by Moosehead Breweries; Geological boundary, a boundary between different geological units; Boundary, software for secure remote access to systems based on trusted identity developed by HashiCorp