Here's the theorem that I have to prove:

A set is called an open set if such that . Prove that the following two statements are equivalent:

(a) \rightarrow R" alt="f \rightarrow R" /> is continuous

(b) For any open set , there exists an open set such that . Here, denotes the preimage of U under f.

Can anyone help me with this? I don't even understand the "idea" of it, much less how to prove it.