Monotone decompositions of $\theta _{n}$-continua

Abstract:

We prove the following theorem for a compact, metric ${\theta _n}$-continuum (i.e., a compact, connected, metric space that is not separated into more than $n$ components by any subcontinuum). The continuum $X$ admits a monotone, upper semicontinuous decomposition $\mathfrak {D}$ such that the elements of $\mathfrak {D}$ have void interiors and the quotient space $X/\mathfrak {D}$ is a finite graph, if and only if, for each nowhere dense subcontinuum $H$ of $X$, the continuum $T(H) = \{ x|$ if $K$ is a subcontinuum of $X$ and $x \in {K^ \circ }$, then $K \cap H \ne \emptyset \}$ is nowhere dense. The elements of the decomposition are characterized in terms of the set function $T$. An example is given showing that the condition that requires $T(x)$ to have void interior for all $x \in X$ is not strong enough to guarantee the decomposition.