# Homework Help: Open and metric

1. Jan 29, 2013

### cummings12332

1. The problem statement, all variables and given/known data

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

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????

2. Jan 29, 2013

### micromass

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

3. Jan 29, 2013

### cummings12332

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)

4. Jan 29, 2013

### micromass

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

5. Jan 29, 2013

### cummings12332

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??

Last edited: Jan 29, 2013