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# Let X be a compact metric space and let U be an open cover of X. Show that there exists a positive real number such that, for every subset E X such...

Let X be a compact metric space and let U be an open cover of X. Show that there exists a positive real number δ such that, for every subset E ⊆ X such that diam(E)<δ, we have E ⊆ u for some u ∈ U.