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## Main Question or Discussion Point

I am trying to understand the theorem:

Let f:S->T be a transformation of the space S into the space T. A necessary and sufficient condition that f be continuous is that if O is any open subset of T, then its inverse image [itex]f^{-1}(O)[/itex] is open in S.

First off, I don't really understand what the inverse image is. Is it just the inverse function? The proof of this theorem deals with the inverse image a lot so I believe that not understanding what an inverse image is, is limiting my ability to understand the proof.

Thanks to anyone who can offer any advice.

Let f:S->T be a transformation of the space S into the space T. A necessary and sufficient condition that f be continuous is that if O is any open subset of T, then its inverse image [itex]f^{-1}(O)[/itex] is open in S.

First off, I don't really understand what the inverse image is. Is it just the inverse function? The proof of this theorem deals with the inverse image a lot so I believe that not understanding what an inverse image is, is limiting my ability to understand the proof.

Thanks to anyone who can offer any advice.