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Homework Help: Open and metric

  1. Jan 29, 2013 #1
    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
    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????
  2. jcsd
  3. Jan 29, 2013 #2
    An obvious first thing to do is to describe the open balls. What do the open balls look like?
  4. Jan 29, 2013 #3
    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)
  5. Jan 29, 2013 #4
    Go on. Can you generalize this to other centers and radii?
  6. Jan 29, 2013 #5
    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
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