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A subset $Y$ of a space $X$ is almost countably compact in $X$ if for every countable cover $𝒰$ of $Y$ by open subsets of $X$, there exists a finite subfamily $𝒱$ of $𝒰$ such that $Y\subseteq \overline{\bigcup 𝒱}$. In this paper we investigate the relationship between almost countably compact spaces and relatively almost countably compact subsets, and also study various properties of relatively almost countably compact subsets.