- #1

tomkoolen

- 40

- 1

Let (X, dx) and (Y,dy) be metric spaces and consider their product topology X x Y (T1) and the topology T2 induced by the metric d((x1,y1),(x2,y2)) = max(dx(x1,x2),dy(y1,y2)) so the maximum of their respective metrics.

I have to show that the topologies are the same. I understand that I have to show that the basis opens of T1 should be unions of basis opens of T2 and vice versa.

T1 => T2 is not giving me a hard time: If (x,y) is in U x V, with U open in X and V open in Y, there exist open balls respectively for x in U and for y in V. The minimum of these radii give a ball for (x,y) in U x V. I know that this is the way and I know how to write it down mathematically correct but I am not sure how to conclude that U x V is in T2 now.

T2 => T1 is giving me more struggles. I can create an open ball around (x0,y0) but then I can't visualise the path my solution should follow.

If someone could give me some advice, I would be very grateful!