I just went through and exercise which asks if a product of two path connected spaces is path connected. There seems no reason to believe it is in general, for the only thing we know is that if two spaces X, Y are path connected, then they are connected, and their product X x Y is connected. So I thought of this Lemma: Let X, Y be two path connected spaces. Their product X x Y is path connected if the domains of the paths fx in X and fy in Y coincide. Assume the hypothesis is true, then for X and Y there exist continuous functions fx : [a, b] --> X and fy : [a, b] --> Y, such that, for any pair of points x1, x2 of X and y1, y2 of Y fx(a) = x1, fx(b) = x2 and fy(a) = y1, fy(b) = y2. Let f : [a, b] --> X x Y be given with f(x) = (fx(x), fy(x)). Since fx and fy are continuous, f is continuous too, and for any pair of points (x1, x2), (y1, y2) of X x Y f(a) = (x1, y1), f(b) = (x2, y2). I think this should work, any comments?