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Let $X$ and $Y$ be metric spaces. Does there exist a metric space $Z$ such that both $X$ and $Y$ can be isometrically embedded into $Z$?

It is easy to see that the answer is positive if $X$ and $Y$ are bounded (consider the disjoint union of the two spaces). But what happens when $X$ or $Y$ is unbounded?

Any help will be appreciated.

jenda358
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    What is going wrong for you with the disjoint union of unbounded spaces? – Rob Arthan Feb 14 '21 at 14:58
  • I don't know how to define a metric on the disjoint union so that the triangle inequality holds. When the spaces are bounded we can define the distance from any point of X to any point of Y to be constant, but this won't work anymore if at least one of the spaces is unbounded. – jenda358 Feb 14 '21 at 17:58
  • With a disjoint union of two metric spaces, how do you define the distance between a point in one space and a point in the other? Obviously in some cases there's more than one way to do it, but how do you show that in every cases there's at least one? – Michael Hardy Feb 14 '21 at 19:02

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Assume $X$ and $Y$ are disjoint. Choose a point $x_0\in X$ and a point $y_0\in Y$. For $x\in X$ and $y\in Y$ define $$d(x,y)=d_X(x,x_0)+1+d_Y(y_0,y).$$

bof
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