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$\operatorname{UC}$ space; boundedly $\operatorname{UC}$ space; boundedly compact space; Atsuji space; uniform continuity on bounded sets; topology of uniform convergence on bounded sets; Attouch--Wets topology
A metric space $\langle X,d\rangle$ is called a $\operatorname{UC}$ space provided each continuous function on $X$ into a metric target space is uniformly continuous. We introduce a class of metric spaces that play, relative to the boundedly compact metric spaces, the same role that $\operatorname{UC}$ spaces play relative to the compact metric spaces.
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