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Thread: Product metric space

  1. #1
    Nov 2008

    Product metric space

    I am trying to show that if U is open in a metric space (X,d) and V is open in (Y,p). Then U \times V is open in the product topology generated by the taxicab metric (X \times Y,e). In order to show this, I tried to prove that B_e((x,y),\epsilon)=B_d(x,\epsilon) \times B_p(y,\epsilon).
    Let (a,b)\in B_e((x,y),\epsilon) <br />
\Leftrightarrow e((a,b),(x,y)) < \epsilon<br />
\Leftrightarrow d(a,x)+p(b,y) < \epsilon<br />
\Leftrightarrow d(x,a) < \epsilon and p(b,y)< \epsilon<br />
\Leftrightarrow a\in B_d(x,\epsilon) and y\in B_p(y,\epsilon)

    However, there is a problem with the 3rd equivalent inequality since it only holds one direction, not the other. I am not sure if I approach this problem correctly. Hope someone can help me on this.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Nov 2010
    Madrid, Spain
    You only need to prove that for every (a,b)\in U\times V there exists an open ball B_e((a,b),\epsilon)\subset U\times V and not necessarily that B_e((a,b),\epsilon)\subset U\times V is the product of two open balls.
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