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Suppose $X$ is a set with a metric $d: X \times X \rightarrow \mathbb{R}$. What "operations" on $d$ will yield a metric in return?

By this I mean a wide variety of things. For example, what functions $g: \mathbb{R} \rightarrow \mathbb{R}$ will make $g \circ d$ into a metric, for example $g \circ d = \sqrt{d}$. Or what functions of metrics will yield metrics in return, for example $d_1 + d_2$, where $d_1$ and $d_2$ are distinct metrics on $X$.

I'm looking for a list of such operations, and counterexamples of ones which plausibly seem like they could define a metric but do not.

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    Of particular interest is finding metric(s) $d'$ that generate the same topology that $d$ does.2018-08-28

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