Solving Problems in French Railway Metric

[cummings12332]

Homework Statement

can someone help me to solve these problems in details??
Consider A =(0,1)× R. Is A open w.r.t. the topology induced by the French railway metric in R2? how
about B=(-1,1)× R?

2. The attempt at a solution
I know A is open in the topology induced by d if and only
if U is a union of metric balls. But for my questions here, how can I see that A is a union of metric balls?? should I take any x in A then exists r>0 s.t. B(x,r) is open in A?but now the French railway metric gives me 2 different cases, how to consider?

 

[micromass]

An obvious first thing to do is to describe the open balls. What do the open balls look like?

 

[cummings12332]

An obvious first thing to do is to describe the open balls. What do the open balls look like?

If I choose the centre be (1,1) and radius 1 to be the open ball then it is segment with length 2 , 45degree to x-axis and the midpoint of the segment is (1,1)

 

[micromass]

If I choose the centre be (1,1) and radius 1 to be the open ball then it is segment with length 2 , 45degree to x-axis and the midpoint of the segment is (1,1)

Go on. Can you generalize this to other centers and radii?

 

[cummings12332]

Go on. Can you generalize this to other centers and radii?

Can I choose the radius of open ball be r=min{1-x1,x1} then for all x=(x1,x2) in A we have B(x,r) is contained in A so A is open??