There aren't a lot of strict prerequisites for topology, but I think in practice it's quite difficult to get anywhere with it or see the point of it without having studied some analysis. A typical progression would be to learn rigorous calculus in a book like Spivak/Apostol/Courant or analysis in a simple book like Burkill, followed by harder analysis in a book like Rudin or Apostol, where you also learn about R\^n and more generally metric spaces, which are a special case of topological spaces. After that a good book to learn topology is the one by Willard.

If you want an introduction to topology for high schoolers with minimal prerequisites, there is a book by Steenrod and Chinn in the "New Mathematical Library". It's more an introduction to what a few parts of the subject are about than a systematic study of topology.