A pseudoMenger space is a set
X together with a function 𝜃;X × X →𝒟, the set of distribution functions,
satisfying certain natural axioms similar to those of a pseudometric space.
Let T : X → X be a bijection and let 𝜃_{r} denote the topology generated by
{T^{i}U(p,𝜖,λ);i ∈ Z,p ∈ X,𝜖 > 0,λ > 0} where U(p,𝜖,λ) = {q;𝜃(p,q)(𝜖) > 1 − λ}.
Assume that 𝜃_{r} is compact. Let h(T,𝜃) denote the topological entropy of T with
respect to the 𝜃_{r} topology. The purpose of this note is to show that if one is given a
sequence {𝜃_{n}} of pseudoMenger structures on X satisfying 𝜃_{n}(p,q) ≧ 𝜃(p,q) and
𝜃_{n}(p,q) → 𝜃(p,q) in distribution for all p,q ∈ X then h(T,𝜃_{n}) → h(T,𝜃). A
counterexample is then given to show that, in general, the condition 𝜃_{n}(p,q) ≧ 𝜃(p,q)
cannot be removed.
