A metrization for power-sets and cartesian products with applications to combinatorial analysis /
Robert D. Silverman · OhioLink ETD Center (Ohio Library and Information Network) · 1958
M 3 .(Symmetry) If • • .>ir is any permutation of the integers 1,2,..., r, then d( a p ..., ar) = d(a^, . .a ^) .M U« (Triangle inequality) For every set of r + 1 points ) «• •» aj.+ ]_ of M, d(a^ *«• >ar) ..»,ar + ^)» The postulate M 2 differs somewhat from that in ELumenthal [U, P* 126], but is more suitable to our purposes.Also, in keeping with the progressive spirit of the times, our "dimension" is one higher than the comparable "dimension" in ELumenthal.In the applications there frequently will be metrics of various dimensions defined over M with connecting conditions.Generally we will use the symbol M to denote either the set M, or the metric space consisting of M and the metric d over M. Wien it is desirable to differentiate explicitly the two concepts, we will symbolize the latter by (M,d).We now define some concepts for conventional (two-dimensional) metric spaces.In most instances the extension to r-dimensional spaces is obvious.In the following, M denotes a metric space, and E a subspace of M.DEFINITION 2. If M ■ £a-^,a2, ...j is countable, it may be completely specified by the symmetric distance matrix A -(ai;j)> ai;j ■ d(a±,a3).