Metrics and entropy for non-compact spaces

We investigate Bowen’s metric definition of topological entropy for homeomorphisms of non-compact spaces. Different equivalent metrics may assign to the homeomorphism different entropies. We show that the infimum of the metric entropies is greater than or equal to the supremum of the measure theoretic entropies. An example shows that it may be strictly greater. If the entropy of the homeomorphism can vary as the metrics vary we see that the supremum is infinity.