Let \(g\) be a polynomial of degree at least two. Denote by \(K(g)\) the filled-in Julia set of \(g;\) that is, \(K(g)=\{z\in \mathbb{C}:g^n(z)\not\to \infty\},\) where \(g^n\) denotes the \(n\)th iterate of \(g.\) The complement of \(K(G)\) in \(\mathbb{C}\) is called the basin of attraction of infinity. A basic result in complex dynamics says that \(\partial K(g)=J(g),\) where \(J(g)\) is the Julia set of \(g.\) This paper is concerned with the question how these results can be extended to the context of polynomial semigroups, the semigroup operation being composition. More specifically, the author considers polynomial semigroups of finite type. These are, by definition, semigroups \(G\) such that for any \(N\) there are at most finitely many polynomials in \(G\) whose degree is less than \(N\) and such that there is a domain \(D_\infty\) in \(\mathbb{C}\) whose complement in \(\mathbb{C}\) is bounded and connected and which satisfies \(g(D_\infty)\subset D_\infty\) for all \(g\in G.\) Aimo Hinkkanen and Gaven Martin [preprint] introduced the set \[ K_1(G)={\overline{\bigcup_{g\in G}g^{-1}\left(\bigcup_{h\in G}K(h)\right)}} \] and showed that the unbounded component \(V\) of \(\mathbb{C} \backslash K_1(G)\) can be considered as the analogue of the immediate basin of attraction of infinity. Here, the author considers the sets \(K(G)\) of all points \(z\) for which \(\{g(z)=g \in G\}\) has a finite limit point, the Julia set \(J(G)\) consisting of all points \(z\) where \(G\) fails to be normal and \(K_2(G)={\overline{\bigcup_{g \in G}K(g)}}.\) He shows that \(V\) is also the unbounded component of the complement of \(K(G),J(G)\) and \(K_2(G),\) and examples are given to show that these inclusions may be strict.