Entropy sequences and maximal entropy sets

The notions of entropy sequences and entropy sets are introduced both in topological and measure-theoretical situations. It turns out that a subset is an entropy set if and only if each n-tuple (not on the diagonal) from the set is an entropy n-tuple and that each maximal entropy set is closed. The systems with only one maximal entropy set are characterized. Moreover, it is proved that if a topological system has positive entropy, then there is a maximal entropy set with uncountably many points, and the topological entropy is the supremum of the entropies over all maximal entropy sets. An example with a maximal entropy set containing only two points is given.