We discuss a technique developed by V. V. Fedorchuk that has been used to solve numerous problems in general topology. Compact spaces constructed using this technique are now widely known as Fedorchuk compacta. Recent interest in this class has emerged in the context of renormings of Banach spaces. A result of the author roughly states that the space of continuous functions on a Fedorchuk compact has good renorming properties as long as the compact satisfies several key requirements. Under additional set-theoretic assumptions we construct an example, showing that the requirements in the theorem are essential.