Intuition on the proposition: $f:(X,d_{X})\rightarrow(Y,d_{Y})$ is continuous iff the pre-image of an open set is open I am studying continuity on metric spaces and I have been presented to the proposition in the title. My question is: how do we understand the kernel of this definition? I am already acquainted to the $\varepsilon-\delta$ definition. I do also know that continuous functions map convergent sequences onto convergent sequences. I am really concerned with the result in the title. 

Fix an $x \in X$. Note that for any $\epsilon > 0$, the ball of radius $\epsilon$ around $f(x)$ is an open set, so the fact that the pre-image of that ball is open implies that you can find a $\delta$-radius ball around $x$ that maps into the $\epsilon$-ball, which is exactly what is happening with the $\epsilon$-$\delta$ definition.
