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Questions of the kind "What's the name for a X that satisfies property Y?"

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Is there a literature name for this concept of a "graded metric"?

Given a space $X$, I have been thinking about a function $d\colon X \times X \times \mathbb{N} \to \mathbb{R}_{\geq 0}$ (i.e. with values that are nonnegative reals) with the properties below. One may …
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