The product theorem for topological entropy
L. Wayne Goodwyn · Transactions of the American Mathematical Society · 1971
The purpose of this paper is to prove the following product theorem : Theorem 2. Let X and Y be compact Hausdorff spaces and let T: X -*■ X and S: Y-*-Y be continuous.Then h(TxS) = h(T) + h(S),where h denotes topological entropy [1], and Tx S: Xx Y-+ Xx Y is defined as Tx S(x, y) = (Tx, Sy) for (x, y)eXxY.This theorem is stated in [1] without the assumption that A'and y be Hausdorff.However, the first half of the proof depends on the assertion that for open covers a of X and ß of Y, N(a xß) = N(a) ■ N(ß).Without much difficulty one can construct examples where this equality does not hold, so that the proof in [1] yields only that h(TxS)^h(T) + h(S).