We solve the longstanding open problems of the blowup involved in the translations (when possible) of a nondeterministic Büchi word automaton (NBW) to a nondeterministic co-Büchi word automaton (NCW) and to a deterministic co-Büchi word automaton (DCW). For the NBW to NCW translation, the currently known upper bound is 2O(n log n) and the lower bound is 1:5n. We improve the upper bound to n2n and describe a matching lower bound of 2(n). For the NBW to DCW translation, the currently known upper bound is 2O(n log n). We improve it to 2O(n which is asymptotically tight. Both of our upper-bound constructions are based on a simple subset construction, do not involve intermediate automata with richer acceptance conditions and can be implemented symbolically. We point to numerous applications of the new constructions. In particular they imply a simple subset-construction based translation (when possible) of LTL to deterministic Büchi word automata.