# Continuous function

Let and be two topological spaces and a function.The function f is said continuous in if and only if for all V a neighborhood of f(x) we have is a neighborhood of x. f is continuous on X if and only if f is continuous in any .
Theorem
Let and be two topological spaces and a function. The following statement are equivalent
(i) f is continuous on X;
(ii) we have ;
(iii) closed in we have is closed in ;
(iv) we have ;
(v) we have

Proof.
Let . means that is open in . But a set is open if and only if the set is a neighborhood for any of its points.
So, let then but is open in so and as f is continuous in x we have and so is a neighborhood for any of its points, so it is open.
Let F be closed in Y. Then is open in Y and by (ii) is open in X, so is closed in X.
Let be . As we have . But from (iii) as is closed in , is closed in and if a set is included in a closed set, its closure is equally included in the same closed set, so ;
Let , then
= or
Let and let , then
or and so so f is continuous in an arbitrary so f is continuous on .