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A dense-in-itself space $X$ is called if the space of real continuous functions on $X$ with its box topology, $C_{\square }\left(X\right)$, is a discrete space. A space $X$ is called provided that $X$ is the union of a countable increasing family of subsets each of them with an empty interior. We analyze these classes of spaces by determining their relations with $\kappa$-resolvable and almost resolvable spaces. We prove that every almost-$\omega$-resolvable space is $C_{\square }$-discrete, and that these classes coincide in the realm of completely regular...