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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 ...
For another example, consider the relative interior of a closed disk in . It is locally closed since it is an intersection of the closed disk and an open ball. On the other hand, { ( x , y ) ∈ R 2 ∣ x ≠ 0 } ∪ { ( 0 , 0 ) } {\displaystyle \{(x,y)\in \mathbb {R} ^{2}\mid x\neq 0\}\cup \{(0,0)\}} is not a locally closed subset of R 2 ...
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 collection of all bounded sets on a topological vector space is called the von Neumann bornology or the (canonical) bornology of .. A base or fundamental system of bounded sets of is a set of bounded subsets of such that every bounded subset of is a subset of some . [1] The set of all bounded subsets of trivially forms a fundamental system of bounded sets of .
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 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.