Abstract. We study effective decomposability of continua in computable metric spaces. A continuum $X$ is called decomposable if there exist proper subcontinua $A,B$ of $X$ such that $X=A\cup B$. We investigate when such a classical decomposition can be replaced by a computable one. We say that a continuum is effectively decomposable if it can be written as the union of two proper computable subcontinua.

If $(X,d,\alpha)$ is itself a continuum, then in order to ask whether decomposability can be made effective, one must first require $X$ to be effectively compact. Even under this assumption, it is not known whether decomposability implies effective decomposability.

Our first result gives one sufficient condition for effective decomposability. Let $(X,d,\alpha)$ be an effectively compact computable metric space such that $(X,d)$ is a continuum. If $X$ contains an open subset homeomorphic to $\mathbb{R}$, then $X$ is effectively decomposable. Consequently, arcs, topological circles and, in general, topological graphs exhibit effective decomposability in this setting.

We also prove a result for chainable continua. If a semicomputable chainable continuum $S$ is decomposable, say $S=K_1\cup K_2$, then we use the fact that $S$ can be inner approximated by a computable subcontinuum $H$. This construction yields computable proper subcontinua $K_1\cup H$ and $K_2\cup H$, and hence an effective decomposition of $S$.

Classically, decomposability is also characterized by the following condition:

\[\textbf{(Ch)}\qquad X \text{ is decomposable if and only if } X \text{ contains a proper subcontinuum with nonempty interior.}\]

Motivated by $\textbf{(Ch)}$, we discuss desirable effective versions of this characterization.