If I have a homeomorphism between $X$ and $Y$. Suppose $X$ is complete and $Y$ is incomplete under the same metric, how can I make $Y$ complete using the fact that it is homeomorphic to $X$. Consider $\mathbb{R}$ and open interval $(0,1)$ how do I make the interval complete under usual metric. Of course if I add $0,1$ to the open interval it will be complete , but what my interest is how do you use the advantage of homeomorphism to make it complete. Thanks in advance.
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In general, if $(X,d)$ is a complete metric space, $Y$ is a topological space and you have a homeomorphism $f:X\to Y$ you can define a metric $d_Y$ on $Y$ which induces the same topology such that $(Y,d_Y)$ is complete. In particular, if you let $d_Y(x,y)=d(f^{-1}(x),f^{-1}(y))$ you get such a metric. |
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Using any homeomorphism $\mathbb R\cong (0,1)$ you can pull back the euclidean metric to get a complete metric on $(0,1)$ inducing the standard topology. Of course, this metric will not be the same as the non-complete metric coming from the canonical embedding of $(0,1)$ into $\mathbb R$. |
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You can't. The issue is that completeness is a purely metric space idea and so should not (and is not, as your example shows) preserved under homeomorphism. |
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