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Let $X$ be a locally connected, $b$-compact metric space and $E$ a closed subset of $X$. Let $𝔾$ be the space of all continuous real-valued functions defined on some closed subsets of $E$. We prove the equivalence of the ${\tau }_{{}_{aw}}$ and ${\tau }_{{}_K}^c$ topologies on $𝔾$, where ${\tau }_{{}_{aw}}$ is the so called topology, defined in terms of uniform convergence of distance functionals, and ${\tau }_{{}_K}^c$ is the topology of Kuratowski convergence on compacta.