I need help with what I am sure just a simple tweek in concept can fix.

In topology, we define a continuous map as follows:

**Definition: **Let

and

be topological spaces. A map

is called continuous if the inverse image of open sets is always open. (says the book, "Topology" by Klaus Janich)

That is, (and this is not in the book)

is a continuous map if for every set

we have

My question is, how can we prove that this definition of continuity is equivalent to the calculus

definition of continuity?

I'm pretty sure I have to say something about "metric spaces" and "neighborhoods" but there is a small conceptual leap i can't make to connect it all.

If necessary, I can supply any definition I am working with upon request, but I don't want someone to write the proof for me, but in layman's terms, tell me how I should think about it, what conceptual links should I be making between calculus and topology in this respect?