Real Analysis Do the closed set consists of points other than limit points? :confused: If yes, then what are the examples? 
Isolated points? 
Yes points that are apart from limit points in the set. I didn't get what you meant by Isolated points. 
Every point $x$ of a set is a limit point of that set (even if it's an "isolated point"). Indeed, consider the constant sequence $x, x, \dots$. 
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Ah yes, ignore my previous post. I mistakenly took $x$ being a limit point to mean "$x$ is the limit of a sequence of points in the set" (rather than "$x$ is the limit of an eventually nonconstant sequence of points in the set"/"every neighbourhood of $x$ contains a point of the set except $x$). 
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An adherent point, also known as a point of closure, is a point whose every neighborhood contains some point of the set. So $2$ is a point of closure of $[0,1] \cup \{2\}$. But it's not a limit point, which requires that every neighborhood of the point contains some point of the set other than the point in question. 
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