# Is the infinite product map $(∏_{i=1}^{∞}S_{i})×f$ topologically transitive

Let $$f:X→X$$ be a map. We say that $$f$$ is topologically mixing if for every open subsets $$U,V$$ of $$X$$, there exists $$N$$ such that for every $$n≥N$$ the set $$f^{n}(U)∩V$$ is non-empty.

Let $$S : X → X$$ and $$T : Y → Y$$ be dynamical systems. Then the map $$S × T$$ is defined by:

$$S × T : X × Y → X × Y$$, $$(S × T)(x, y) = (Sx, Ty)$$.

We know that if $$S$$ and $$T$$ are topologically transitive and at least one of them is mixing then $$S × T$$ is topologically transitive (https://www.merry.io/dynamical-systems-lecture-notes/2016/10/3/the-shift-map).

My question is: Consider an infinite set of mixing maps $$S_{i}:X_{i}→X_{i}$$ along a topologically transitive map $$f$$.

Is the infinite product map $$(∏_{i=1}^{∞}S_{i})×f$$ topologically transitive.