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A cover $\mathcal C$ of a uniform space $(X,\mathcal U)$ is called a uniform cover if there is $U\in \mathcal U$ such that the cover $\{U(x):x\in X\}$ refines $\mathcal C$.

Is it true that to every uniform cover of a uniform space one can subordinate an equiuniformly continuous partition of unity? Is this true for any uniform space or some condition has to be taken like that of paracompactness?

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