http://www.mathcs.org/analysis/reals/cont/proofs/conttop.htmlFor the proof (2) => (1).

(2) => (1): Assume that the inverse image f-1(U) of every open set U is open.

Take any point x0 (belong to ) "U" and pick an ε > 0.

Should it be : take any point x0 (belong to) "D", insteand belong to "U"?