# A-system

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A "countably-ramified" system of sets, i.e. a family of subsets of a set , indexed by all finite sequences of natural numbers. An -system is called regular if . A sequence of elements of an -system indexed by all segments of one and the same finite sequence of natural numbers is called a chain of this -system. The intersection of all elements of a chain is called its kernel, and the union of all kernels of all chains of an -system is called the kernel of this -system, or the result of the -operation applied to this -system, or the -set generated by this -system. Every -system can be regularized without changing the kernel (it suffices to put ). If is a system of sets, then the kernels of the -system composed from the elements of are called the -sets generated by . The -sets generated by the closed sets of a topological space are called the -sets of this space.

#### References

 [1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian) [2] K. Kuratowski, "Topology" , 1 , Acad. Press (1966) (Translated from French)