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Is it correct to say that this set $E=(0,1]$ where $E\subseteq R$ (Where $R$ is the set of real numbers) is not closed?

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Yes. Closed sets are the complements of open sets. Therefore we check if the complement of $(0,1]$ is open or not. $$\mathbb{R}\backslash(0,1]=(-\infty,0] \cup (1,\infty)$$ An open set is a set for all points there exists a neighborhood contained in the set with positive radius. But, for point 0, no matter how little you go to $+$ side, you always go off the set. Therefore $\mathbb{R}\\(0,1]$ is a not open set, and therefore $(0,1]$ is a not closed set.