Monotone decompositions of $\theta$-continua

Abstract:

A $\theta$-continuum (${\theta _n}$-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 ${\theta _n}$-continua, and the second generalizes earlier work by Vought for ${\theta _1}$-continua. A $\theta$-continuum $X$ admits a monotone, upper semicontinuous decomposition $\mathcal {D}$ such that the elements of $\mathcal {D}$ have void interiors and the quotient space $X/\mathcal {D}$ is a finite graph, if and only if, for each nowhere dense subcontinuum $H$ of $X$, the continuum $T(H) = \{x \in X|$ if $K$ is a subcontinuum of $X$ and $x$ is in the interior of $K$, then $K \cap H \ne \emptyset \}$ is nowhere dense. Also, if $X$ satisfies this condition, then $X$ is in fact a ${\theta _n}$-continuum, for some natural number $n$, and, for each natural number $m$, $X$ is a ${\theta _m}$-continuum, if and only if $X/\mathcal {D}$ is a ${\theta _m}$-continuum.