Based on the results from (Mihail and Miculescu in Math. Rep., Bucur. 11(61)(1):21-32, 2009), where the shift space for an infinite iterated function system (IIFS for short) is defined and the relation between this space and the attractor of the IIFS is described, we give a sufficient condition on a family ( I j ) j ∈ L of nonempty subsets of I, where S = ( X , ( f i ) i ∈ I ) is an IIFS, in order to have the equality ? j ∈ L A I j ˉ = A , where A means the attractor of S and A I j means the attractor of the sub-iterated function system S I j = ( X , ( f i ) i ∈ I j ) of S . In addition, we prove that given an arbitrary infinite cardinal number A , if the attractor of an IIFS S = ( X , ( f i ) i ∈ I ) is of type A (this means that there exists a dense subset of it having the cardinal less than or equal to A ), where ( X , d ) is a complete metric space, then there exists S J = ( X , ( f i ) i ∈ J ) a sub-iterated function system of S , having the property that card ( J ) ≤ A , such that the attractors of S and S J coincide. MSC:28A80, 54H25.
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机译：根据（Mihail and Miculescu in Math。Rep。，Bucur。11（61）（1）：21-32，2009）中的结果，其中定义了无限迭代函数系统（简称IIFS）的移位空间并描述了该空间与IIFS吸引子之间的关系，我们给出了I的非空子集的族（I j）j∈L的充分条件，其中S =（X，（fi）i∈I）是一个IIFS，为了拥有平等？ j∈L A I jˉ= A，其中A表示S的吸引子，A I j表示S的子迭代函数系统S I j =（X，（f i）i∈I j）的吸引子。另外，我们证明给定任意基数A，如果IIFS S =（X，（fi）i∈I）的吸引子为A型（这意味着存在具有基数的密集子集）小于或等于A），其中（X，d）是一个完整的度量空间，则存在SJ =（X，（fi）i∈J）的S的子迭代函数系统，其属性为card（ J）≤A，使得S和SJ的吸引子重合。 MSC：28A80，54H25。
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