this is what i did for continuity im not sure why the empty set is open

f is continuous: Suppose U X V C X X Y is the basis element of the product topology (by the definition of product topology on page 86 in the textbook). Then by the definition of product topology since U X V C X X Y then, U C X and V C Y are both open subspaces. Thus, the inverse image of U X V of f is: 23 Peter Krahn MATH-T 620: Topics in Topology/Geometry Topic: Topology 1 Dr. Jim Carter Midterm Exam f -1 (U X V) = {x EX| (x, yo) EUXV} f - 1 ( U X V ) = {x EX| x EU, yo EV} Thus, When yo E V then, f-1(U X V) = U where U is open . When yo # V then, f-1(U X V) = 0 which is open