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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 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.
3. Between two groups, may mean that the first one is a proper subgroup of the second one. > (greater-than sign) 1. Strict inequality between two numbers; means and is read as "greater than". 2. Commonly used for denoting any strict order. 3. Between two groups, may mean that the second one is a proper subgroup of the first one. ≤ 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 .
The boundary of a cell is the system of edges that touch it, and the boundary of an edge is the set of vertices that touch it (one vertex for a ray and two for a line segment). The system of objects of all three types, linked by this boundary operator, form a cell complex covering the plane.
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).
An exterior domain or external domain is a domain whose complement is bounded; sometimes smoothness conditions are imposed on its boundary. In complex analysis , a complex domain (or simply domain ) is any connected open subset of the complex plane C .
An artist's impression of a bounded set (top) and of an unbounded set (bottom). The set at the bottom continues forever towards the right. In mathematical analysis and related areas of mathematics, a set is called bounded if all of its points are within a certain distance of each other.