Let $X$ be a separable topological space of size $\mathfrak{c}$. By a simple function $\phi:X\to X$, we mean a finite range valued measurable function. 


Q. Is it possible to find a sequence of simple functions $\{\phi\_n\}$ with $\lim\phi\_n(x)=x$ for every $x\in X$?



Approximation on separable topological space with size $\mathfrak{c}$
No, take any $\mathbb{N} \subseteq X \subseteq \beta\mathbb{N}$ with $|X|=\mathfrak{c}$. If ${\phi\_n}$ is any sequence of functions with finite range (measurable or not) and $x\in X \setminus \bigcup\_n \phi\_n(X)$, then $\lim \phi\_n(x) \neq x$.


