Compact metric spaces without continuous surjective functions between them.

Question

There exist two compact metric spaces such that doesn't exist a continuous and surjective function between them? That is, two compact metric spaces $X$ and $Y$ such that for every surjective functions $f:X \to Y$ and $g:Y \to X$, $f$ and $g$ aren't continuous.

Answer 1

Let $X=\{0\}\cup\left\{\frac1n:n\in\Bbb Z^+\right\}$ and $Y=[0,1]$, both with the usual metric inherited from $\Bbb R$; both are compact metric spaces. There is no surjection at all from $X$ to $Y$, since $X$ is countable and $Y$ is uncountable, and there is no continuous surjection from $Y$ to $X$, since $Y$ is connected, and $X$ is not. (Recall that continuous functions preserve connectedness.)

Answer 2

This is not possible. Let $X, Y$ be non-empty, then there exists a constant function from $X$ to $Y$ and one in the reverse direction. Both are continuous, since constant functions are continuous.

If one of them is empty, the functions between $X$ and $Y$ are all continuous (vacuously).