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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.
In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [1] As an example, " is less than " is a relation on the set of natural numbers ; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3 ), and likewise between 3 and 4 (denoted as 3 < 4 ), but not between the ...
The definitions can be generalized to functions and even to sets of functions. Given a function f with domain D and a preordered set (K, ≤) as codomain, an element y of K is an upper bound of f if y ≥ f (x) for each x in D. The upper bound is called sharp if equality holds for at least one value of x. It indicates that the constraint is ...
MV Boundary, a number of ships with this name; Boundaries in landscape history, the divide between areas of differing land used; Boundary (real estate), the legal boundary between units of real property; Boundary (company), an American application performance management company; Boundary critique, a concept about the meaning and validity of ...
a touches b, they have at least one boundary point in common, but no interior points. Contains: a ∩ b = b: Covers: a ο ∩ b = b b lies in the interior of a (extends Contains). Other definitions: "no points of b lie in the exterior of a", or "Every point of b is a point of (the interior of) a". CoveredBy Covers(b,a) Within: a ∩ b = a
These relationships can also be classified semantically: Inherent relationships are those that are important to the existence or identity of one or both of the related phenomena, such as one expressed in a boundary definition or being a manifestation of a mereological relationship.
If, for some notion of substructure, objects are substructures of themselves (that is, the relationship is reflexive), then the qualification proper requires the objects to be different. For example, a proper subset of a set S is a subset of S that is different from S , and a proper divisor of a number n is a divisor of n that is different from n .