Could someone specify a metric on (0,1)

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The discussion focuses on defining a metric on the open interval (0,1) that maintains the same topology as the absolute value metric while ensuring completeness. A homeomorphism T: (0,1) → ℝ is utilized to pull back the metric from ℝ. The proposed metric is defined as d(x,y) = |T(x) - T(y)|, effectively transforming the interval into a complete metric space.

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could someone specify a metric on (0,1) that defines (the same topology) as the abs. value (i.e. usual) metric and makes this open interval into a complete set?

Thanks
 
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There is a homeomorphism [itex]T:(0,1)\rightarrow \mathbb{R}[/itex]. Pull back the metric from [itex]\mathbb{R}[/itex]. Thus define

[tex]d(x,y)= |T(x)-T(y)|[/tex]
 

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