engleski

Computable transition points

Suppose $(X, d, \alpha)$ is a computable metric space. Let $U$ and $V$ be disjoint and computably enumerable open sets and let $K$ be a computable compact set in this space. Let $S=X\setminus (U\cup V)$. Suppose $K$ is a continuum chainable from $a$ to $b$, where $a$ and $b$ are computable points, $a\in U$, $b\in V$. We prove the following: if $K\cap S$ is totally disconnected, then $K\cap S$ contains a computable point.

computable metric space ; computably enumerable open set ; computable compact set ; computable point ; chainable continuum

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