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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.
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right.
Some illustrative examples of non-orientable manifolds include: (1) the Möbius strip, which is a manifold with boundary, (2) the Klein bottle, which must intersect itself in its 3-space representation, and (3) the real projective plane, which arises naturally in geometry.
Genus (mathematics) Examples Positive Euler characteristic. 2-disk; Sphere; Real projective plane; Zero Euler characteristic Annulus; Möbius strip; Torus; Klein bottle; Negative Euler characteristic The boundary of the pretzel is a genus three surface; Embedded/Immersed in Euclidean space Cross-cap; Boy's surface; Roman surface; Steiner ...
The following Application Integrated Resources (AICs) define boundary models that are constraints of the generic geometric and topological capabilities: ISO 10303-511 Topologically bounded surface , definition of an advanced face , that is a bounded surface where the surface is of type elementary (plane, cylindrical, conical, spherical or ...
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups can act non-trivially (that is, when the groups in question are realized as geometric symmetries or ...
For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak. Vertex figure: not itself an element of a polytope, but a diagram showing how the elements meet.
For example, in his influential monographs on elliptic partial differential equations, Carlo Miranda uses the term "region" to identify an open connected set, [10] [11] and reserves the term "domain" to identify an internally connected, [12] perfect set, each point of which is an accumulation point of interior points, [10] following his former ...
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