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Theorem (Sierpinski): Let f_1,f_2 ... be any countable set of maps from an infinite set E to itself. Then there exist two functions A and B: E\to E, such that any f_i can be represented as a finite superposition of A and B.
Proof (Banach): Choose a partition of E into countable union of subsets of cardinality |E|:
E_0, E_1, ...
Furthermore, let us partition E_0 as a countable union of subsets of cardinality |E|:
E_01, E_02, ...
Define A as a function on E that sends each E_n to E_n+1 bijectively
Let B outside E_0 be any function that sends bijectively E_n to E_0n
and on E_0n let's define it by
f_n(BA^nBA)^-1
note that it is well defined since BA^nBA maps bijectively E to E_0n
Now f_n=BBA^nBA
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