Abstract:
The problem whether the topological entropy of a continuous self-mapping on a topological space is invariant when the space becomes smaller or its topology becomes weaker is studied.Relevant properties of the topological entropy of a continuous self-mapping on a topological space are discussed by applying the theory of open cover, invariant subset and non-wandering set. Based on this, it is proved that if non-wandering set of the continuous self-mapping f on a topological space is compact, then the topological entropy of f is the same as that of the restriction of f to its non-wandering set, and the topological entropy of a continuous self-mapping on a strong θ-compact Hausdorff space (X,T) is the same as that of the mapping on (X,θ(T)),where θ(T) is the set of all θ-open sets in (X,T).
