Monotone Decompositions of «continua1

A 0-continuum (9„-continuum) is a compact, connected, metric space that is not separated into infinitely many (more than n) components by any subcontinuum. The following results are among those proved. The first generalizes earlier joint work with E. J. Vought for ^„-continua, and the second generalizes earlier work by Vought for 6,-continua. A 0-continuum X admits a monotone, upper semicontinuous decomposition <>D such that the elements of ^D have void interiors and the quotient space X/tf) is a finite graph, if and only if, for each nowhere dense subcontinuum H of X, the continuum T(H) = [x e X\ if K is a subcontinuum of X and x is in the interior of K, then K n H =£ 0 } is nowhere dense. Also, if X satisfies this condition, then X is in fact a ^„-continuum, for some natural number n, and, for each natural number m, AT is a (^-continuum, if and only if X/6!) is a 0m-continuum.