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Ideals which generalize \((v^{0})\). (English) Zbl 1222.03050

A trimmed tree is defined by the authors as follows. Given a countable product of finite sets with more than one point, \(X=\prod_{i\in\omega}X_{i}\), fix \(x\in\prod_{i\in\omega}X_{i}\) and an infinite subset \(A\subseteq\omega\); then define \(T\left[ A,x,0\right] =\) \(\left\{ \left\{ \left\langle 0,x_{0}\right\rangle \right\} \right\} \) whenever \(0\notin A\) and \(T\left[ A,x,0\right] =\) \(\left\{ \left\{ \left\langle 0,t\right\rangle \right\} :t\in X_{0}\right\} \) for \(0\in A\). Then recursively define \[ T\left[ A,x,n\right] =\left\{ s\cup\left\langle n,x_{n}\right\rangle :s\in T\left[ A,x,n-1\right] \right\} \] whenever \(n\notin A\), and \[ T\left[ A,x,n\right] =\left\{ s\cup\left\{ \left\langle n,t\right\rangle \right\} :s\in T\left[ A,x,n-1\right] \wedge t\in X_{n}\right\} \] for \(n\in A\). Then the trimmed tree is \(T\left[ A,x\right] =\left\{ \emptyset\right\} \cup\bigcup\left\{ T\left[ A,x,n\right] :n\in \omega\right\} \). When each set \(X_{n}\), \(n\in\omega\), is considered with the discrete topology, the subset \(\left[ T\left[ A,x\right] \right] =\left\{ y\in X:y\upharpoonright\omega\setminus A=x\upharpoonright\omega\setminus A\right\} \) is a perfect subset of \(X\) equipped with the product topology. The authors consider the families \(\mathcal{V}=\left\{ \left[ T\left[ A,x\right] \right] :A\in\left[ \omega\right] ^{\omega}\wedge x\in X\right\} \) and \(\mathcal{V}^{\ast}=\left\{ \left[ T\left[ A,x\right] \right] ^{\ast}:\left[ T\left[ A,x\right] \right] \in\mathcal{V}\right\} \), where \[ \left[ T\left[ A,x\right] \right] ^{\ast}=\left\{ y\in X:\left( \exists n\in\omega\right) \left( y\upharpoonright\omega\setminus\left( A\cup n\right) \right) =x\upharpoonright\omega\setminus\left( A\cup n\right) \right\} . \] They also define, for a family \(\mathcal{F}\) of subsets of \(X\), the ideal \[ d^{0}( \mathcal{F}) =\left\{ Y\subseteq X:\left( \forall V\in\mathcal{F}\right) \left( \exists U\in\mathcal{F}\right) \left( U\subseteq V\wedge U\cap Y=\emptyset\right) \right\} . \] The ideal \(( v^{0}) \) in the title of the paper is the ideal \(d^{0}( \mathcal{V}) \) when \(X_{n}=\left\{ 0,1\right\} \) for every \(n\in\omega\). The properties of such an ideal were considered before, and in the present paper the authors generalize those considerations to the \(\sigma\)-ideal \(d^{0}( \mathcal{V}) =d^{0}( \mathcal{V}^{\ast})\). The equality is a consequence of the fact that decreasing sequences of elements of \(\mathcal{V}^{\ast}\) are bounded, and proving that it is a \(\sigma\)-ideal requires some fusion relations that the authors present in the paper. They also show a version of the famous Base Matrix Tree Theorem of Balcar, Pelant and Simon in this context. This result is interesting in several ways. Finally, they also compute some cardinal invariants of the ideal.

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
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References:

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