## Combining metrics [closed]

I have two metric spaces, say $\left(X,d_X\right)$ and $\left(Y,d_Y\right)$. My problem is quite simple: I need to combine metrics $d_X$ and $d_Y$ to form metric $d_{XY}\colon\left(X\times Y\right)\times\left(X\times Y\right)\to\mathbb{R}$ such that $\left(X\times Y,d_{XY}\right)$ is also a metric space. What is the simplest way to do that?

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 I don't think this is MO level. Moreover, it is not a well-posed question. Indeed often the right definition depends on what you want to do. Example: if $X,Y$ were connected graph, then you would want the $\ell^1$ metric on the product; in many other cases you would want $\ell^2$.. – Valerio Capraro Jan 31 2012 at 0:07 Voting to close as off topic. $\;$ See mathoverflow.net/faq#whatnot $\;\;$ – Ricky Demer Jan 31 2012 at 0:10