Continuous image of a cantor set and other space filling curves

Question

I am studying the theory of space filling curves, more specifically looking at the continuous image of a Cantor set.

I have been given this definition to characterise the continuous image of a Cantor set.

"Every compact set is the continuous image of the Cantor set".

Does this refer to the compact set in the topological sense or the metric space sense?

In general when we are referring to the continuous image of a space filling curve, are we talking about them in the topological space or the metric space?

So would the following theorem (below) be valid in the theory of space filling curves?

Let $f$ be a function $f:X \to Y$ where $X$ and $Y$ are topological spaces. The function $f$ is continuous on X if and only if the pre-image of $f(V)$, where $V$ is a neighbourhood, is open in $X$ whenever $V$ is open in $Y$

I have learnt from the comments below that there is no difference between compact sets in the topological sense and compact sets in metric sense.

What about other topological concepts like connectedness etc...?

Answer 1

The correct statement is that every nonempty compact metrizable space is a continuous image of the ternary Cantor set (Alexandroff-Hausdorff theorem).