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In this sense interior and closure are dual notions. The exterior of a set S is the complement of the closure of S; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty).
Each clopen subset of (which includes and itself) is simultaneously a regular open subset and regular closed subset. The interior of a closed subset of is a regular open subset of and likewise, the closure of an open subset of is a regular closed subset of . [2] The intersection (but not necessarily the union) of two regular open sets is a ...
The surface S is said to be boundary-compressible if either S is a disk that cobounds a ball with a disk in or there exists a boundary-compressing disk for S in M. Otherwise, S is boundary-incompressible. Alternatively, one can relax this definition by dropping the requirement that the surface be properly embedded.
A set (in light blue) and its boundary (in dark blue). 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 boundary of the body is of zero volume. Thus, the volume of the body is the volume of its interior, and the interior can be exhausted by an infinite sequence of cubes. In contrast, the boundary of an arbitrary set of points can be of non-zero volume (an example: the set of all rational points inside a given cube).
The interior of a line segment in an at least two-dimensional ambient space is empty, but its relative interior is the line segment without its endpoints. The interior of a disc in an at least three-dimensional ambient space is empty, but its relative interior is the same disc without its circular edge.
In the following, represents the real numbers with their usual topology. The subspace topology of the natural numbers, as a subspace of , is the discrete topology.; The rational numbers considered as a subspace of do not have the discrete topology ({0} for example is not an open set in because there is no open subset of whose intersection with can result in only the singleton {0}).
The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre. Meagre sets play an important role in the formulation of the notion of Baire space and of the Baire category theorem , which is used in the proof of several fundamental results of ...