# A-system

A "countably-ramified" system of sets, i.e. a family $\{ A _ {n _ {1} \dots n _ {k} } \}$ of subsets of a set $X$, indexed by all finite sequences of natural numbers. An $A$- system $\{ A _ {n _ {1} \dots n _ {k} } \}$ is called regular if $A _ {n _ {1} \dots n _ {k} n _ {k+1} } \subset A _ {n _ {1} \dots n _ {k} }$. A sequence $A _ {n _ {1} } \dots A _ {n _ {1} \dots n _ {k} } \dots$ of elements of an $A$- system indexed by all segments of one and the same finite sequence of natural numbers is called a chain of this $A$- system. The intersection of all elements of a chain is called its kernel, and the union of all kernels of all chains of an $A$- system is called the kernel of this $A$- system, or the result of the ${\mathcal A}$- operation applied to this $A$- system, or the ${\mathcal A}$- set generated by this $A$- system. Every $A$- system can be regularized without changing the kernel (it suffices to put $A _ {n _ {1} \dots n _ {k} } ^ \prime = A _ {1} \cap \dots \cap A _ {n _ {1} \dots n _ {k} }$). If ${\mathcal M}$ is a system of sets, then the kernels of the $A$- system composed from the elements of ${\mathcal M}$ are called the ${\mathcal A}$- sets generated by ${\mathcal M}$. The ${\mathcal A}$- sets generated by the closed sets of a topological space are called the ${\mathcal A}$- sets of this space.