Every compact metric space is image of space $2^{\mathbb{N}}$

Question

Proposition: Every non empty compact metric space is image of space $2^{\mathbb{N}}$, the cartesian product of countably infinite copies of two point space i.e. for every compact metric set $X$ there exists a continuous onto mapping $$h:2^{\mathbb{N}}\to X$$ where $2^{\mathbb{N}}=\displaystyle\prod_{n=1}^{\infty}X_n$ ; $X_n=\{a,b\}$ for all n. Now, can anyone please help me in finding out any example?

Answer 1

Let $Y$ be a nonempty compact metric space. Then, for some $n_1\in\mathbb N$, $Y$ is the union of $n_1$ nonempty closed sets of diameter $\lt1.$ Next, for some $n_2\in\mathbb N$, each of those $n_1$ sets is the union of $n_2$ (not necessarily distinct) nonempty closed sets of diameter $\lt1/2.$ Next, for some $n_3\in\mathbb N$, each of the previously chosen $n_1n_2$ sets is the union of $n_3$ nonempty closed sets of diameter $\lt1/3$. And so on. Use these coverings in the obvious way to define a continuous surjection from the infinite product space $X=D(n_1)\times D(n_2)\times\cdots$ to $Y,$ where $D(n)$ is a discrete space of cardinality $n.$ Finally, observe that $X$ is a continuous image (in fact a homeomorph but we don't need that) of the Cantor space $2^\mathbb N.$