Proposition 2.30. Suppose X and Y are topological spaces, and f : X → Y is any map. (a) f is continuous if and only if f (A c f(A) for all ASX. (b) f is closed if and only if f (A) 2 f(A) for all A CX. (c) f is continuous if and only if f -(Int B) Intf-?(B) for all B CY. (d) f is open if and only if f-'(Int B) 2 Int f-'(B) for all B C Y.