Originally Posted by

**Jhevon** 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 $\displaystyle X$ and $\displaystyle Y$ be topological spaces. A map $\displaystyle f: X \to Y$ 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) $\displaystyle f: (X_1, Y_1) \to (X_2,Y_2)$ is a continuous map if for every set $\displaystyle U \in Y_2$ we have $\displaystyle f^{-1}(U) \in Y_1$

My question is, how can we prove that this definition of continuity is equivalent to the calculus $\displaystyle \epsilon - \delta$ 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?