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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.
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 ...
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.
After a line, a circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Considering, for instance, the top part of the unit circle, x 2 + y 2 = 1, where the y-coordinate is positive (indicated by the yellow arc in Figure 1).
The edge boundary is the set of edges with one endpoint in the inner boundary and one endpoint in the outer boundary. [ 1 ] These boundaries and their sizes are particularly relevant for isoperimetric problems in graphs , separator theorems , minimum cuts , expander graphs , and percolation theory .
A closed set contains its own boundary. In other words, if you are "outside" a closed set, you may move a small amount in any direction and still stay outside the set. This is also true if the boundary is the empty set, e.g. in the metric space of rational numbers, for the set of numbers of which the square is less than
When two cells in the Voronoi diagram share a boundary, it is a line segment, ray, or line, consisting of all the points in the plane that are equidistant to their two nearest sites. The vertices of the diagram, where three or more of these boundaries meet, are the points that have three or more equally distant nearest sites.
The boundary of a cross-section in three-dimensional space that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space ...