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I was going through real analysis book of savita atora... there was a definition of closure and closed. It was given that the set of all adherent points is called closure of a set that is the the union of derived set and the set itself While the set is called closed if it contains all its limit point So what is the difference between both of them. Is the set $$\left(\frac{1}{n}\right)_{n=1}^{\infty} \bigcup \{0\} $$ is closed or closure

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  • $\begingroup$ Being closed is a topological attribute of a set itself. A set's closure is the smallest set that is closed and contains the original set. Also, please make your notation in order. $\endgroup$ – Vim May 25 '17 at 7:30
  • $\begingroup$ Closure of a set A is a smallest closed set $\bar{A}$ which contains A. The set you gave is a closed set because it contains all of limit points (particularly zero point), it is also the closure of the set $\{1,1/2,1/3,...\}$ $\endgroup$ – Dynamic May 25 '17 at 7:31
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A set $A$ can only be referred to as a closure if mentioned alongside the set that we are taking the closure of. So its not likely that you will ever read that "the set $A$ is closure." in a topology book

A set $A$ is closed if it has the property that it already contains all of its limit points, that is it is equal to its own closure.

For example the set $\{1/n\}_{n=1}^\infty\cup \{0\}$ (assumed to be a subset of $\mathbb{R}$ equipped with the standard topology) is the closure of the set $\{1/n\}_{n=1}^\infty$ as $0$ is the only limit point.

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