topological entropy is self mapping

Question

As I know, topological dynamical system (with discrete time) is a self mapping such that $f:X\rightarrow X$ for $X$ be any arbitrary topological space. But, can we consider mapping

1. $f:N\rightarrow R$ (as $N$ is natural number and $R$ is real number)

or

2. $f:[0,1]\rightarrow (0,1]$

as topological dynamical system with discrete time?

Answer 1

To answer your question, ask yourself: do the iterated compositions of $f$ with itself make sense? In other words, are these all defined: $$f \circ f, \quad f \circ f \circ f, \quad f \circ f \circ f \circ f, \quad \ldots \quad \text{and, by induction,} \quad f^n = f^{n-1} \circ f $$ For a function $f : A \to B$, this is possible if $B \subset A$, whereas if $B \not\subset A$ then it may be impossible.

So for question 2, since $(0,1] \subset [0,1]$ then yes, it is a dynamical system.

And for question 1, since $R \not\subset N$ then no, it need not be a dynamical system.