On the Boundaries of Sets and Boundaries of Convex Sets
УДК 515.12: 514.172
Abstract
The article studies the conditions for the coincidence of the boundaries of a set, its closure, and its interior. A connection is revealed between the coincidence of the boundaries of a set with the boundary of its closure (interior) and the coincidence of the boundaries of the complement of this set with the boundary of its interior (closure). A formula for the boundary of a set boundary in an arbitrary topological space is obtained.
The main results are the following:
- The boundary of a boundary of a set is the union of the boundaries of its interior and closure.
- In order for the boundary of a set to coincide with the boundary of its interior or the boundary of its closure, it is necessary (but not sufficient) that the interior of the boundary of this set be empty.
- The boundary of a convex body in an n-dimensional affine space An coincides with the boundary of its closure and the boundary of its interior. The complement of a convex set has the same property. An example of a star set that does not have this property is given.
The proof methods are topological and also use facts from the theory of convex sets.
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References
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