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A relation R on a set S is transitive provided that for all x, y and z in S such that xRy and yRz, we also have xRz.

Let D be a set of positive numbers containing 1, then the D-distance graph X(D) on a nonempty subset X of Euclidean space is the graph with vertex set X and edge set ...

A set which contains a nonnegative integral number of elements is said to be finite. A set which is not finite is said to be infinite. A finite or countably infinite set is ...

A subset E of a topological space S is said to be of first category in S if E can be written as the countable union of subsets which are nowhere dense in S, i.e., if E is ...

An infinite set, such as the real numbers, which is not countably infinite.

A subset is a portion of a set. B is a subset of A (written B subset= A) iff every member of B is a member of A. If B is a proper subset of A (i.e., a subset other than the ...

A distance graph with distance set (0,1].

The function frac(x) giving the fractional (noninteger) part of a real number x. The symbol {x} is sometimes used instead of frac(x) (Graham et al. 1994, p. 70; Havil 2003, ...

The space called L^infty (ell-infinity) generalizes the L-p-spaces to p=infty. No integration is used to define them, and instead, the norm on L^infty is given by the ...

A mathematical object (such as a set or function) is said to bounded if it possesses a bound, i.e., a value which all members of the set, functions, etc., are less than.