Understanding the product topology

[tomkoolen]

I am having some trouble visualising the following problem and I hope someone will be able to help me:

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!

 

[micromass]

What do open balls in $(X\times Y,d)$ look like? For example, what do the open balls look like in $\mathbb{R}^2$?

 

[tomkoolen]

Thanks for replying! In R2 I could just visualise them as circles around a point (x,y) right?

 

[micromass]

No, not with this metric.

 

[tomkoolen]

I really don't see it. Of course the metric maps R x R ---> R, for example the distance between (1,2) and (2,4) would be 2 but I don't know how to "draw" it. This is the part where I need a hint.

 

[tomkoolen]

Oh I think I should think about the set of all elements of R2 that have fixed distance from (x0,y0) and draw that right?

 

[micromass]

What are the points a distance of 1 away from (0,0)?

 

[tomkoolen]

{(a,1), (1,b), (c,-1), (-1,d)} with a,b,c,d smaller or equal to 1?
ABsolute value!

 

[micromass]

So what figure does that form if you draw it?

 

[tomkoolen]

Two line segments of length 2, a "cross"