Partition of bounded set on finite family of subsets with diameter less then 1

Question

Let $(X,d)$ be a metric space with bounded metric $d$ which can take arbitrary small positive values. I wish to divide $X$ on the finite union of subsets with diameter $<1$? How to do it?

Answer 1

It's not true. Start with any metric space $(Y,d_Y)$ with bounded metric $d_Y$ which can take arbitrarily small positive values. Define $X$ to be a disjoint union of infinitely many copies $Y_1,Y_2,…$ of $Y$. Define the metric $d$ on $X$ so that the restriction of $d$ to $Y_i$ is the same as the metric $d_Y$, and so that $d(y,y')=1$ for all $y \in Y_i$, $y' \in Y_j$, $i \ne j$. It follows that each nonempty subset of diameter $<1$ intersects exactly one $Y_i$, so infinitely many such subsets are needed to cover $X$.