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(or covering), a collection of point sets (geometric figures) whose union forms or contains a given set (or given figure). For example, the diagonal of a rectangle divides it into two triangles that form a cover of the given rectangle. Finite covers—that is, covers consisting of a finite number of elements —are usually considered. If the diameter of each of the sets of a cover is smaller than a given positive ∊, the cover is called an ∊-cover.
For any ∊ > 0, a bounded region permits of a finite ∊-cover by closed sets with at most three sets intersecting at a time, but it permits of no such cover, for a sufficiently small ∊, with only two sets intersecting at a time. Thus, a town square can be covered with arbitrarily small paving blocks in such a way that the stones in the pavement will border only in threes; borderings in threes cannot be avoided. Similarly, when a space is filled with brickwork, the bricks can be made to border only in fours; the borderings cannot be only in threes. Hence, the importance of the concept of the multiplicity of a cover. We say that the multiplicity of a cover of a given set does not exceed n if every point of the set belongs to no more than n sets of the cover. Thus, the multiplicity of finite covers makes it possible to characterize the number of dimensions of a space. In topology, covers are a powerful means of investigating various geometric properties of sets.
P. S. ALEKSANDROV
ii. The protection of friendly forces (ground, maritime, or air) by friendly aircraft and/or by EW (electronic warfare) aircraft.
iii. That which conceals or protects, such as darkness, a topographical feature, a bank of clouds, a shell barrage, or a deceptive move.