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) ) for all b cy. (d) f is open if and only if f-'(int b) 2 int f-'(b) for all b c y.