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Let E be a set. Two definitions of closed sets in literature are:

  1. E is closed if E contains the limit of every convergent sequence of points in E. [Ross, 2013]

  2. E is closed if E contains every limit point of E. [Rudin, 1976]

The set of limits of every convergent sequence of points in E, and the set of limit points of E, is not necessarily equal, so are these two definitions equivalent?

TSJ
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    They are equivalent in metric spaces (and in fact in a considerably larger class of spaces), but they are not equivalent in topological spaces in general. – Brian M. Scott Feb 24 '15 at 10:30
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    Both spaces in this answer are examples of spaces in which the two definitions are not equivalent: in each case the point of $X$ sent to $1$ is in the closure of the rest of the space but not the limit of any sequence in the rest of the space. – Brian M. Scott Feb 24 '15 at 10:38
  • Interesting - let's consider metric spaces only then. Let $A$ denote the set of limits of every convergent sequence of points in $E$, and let $B$ denote the set of limit points of $E$. As mentioned, $A$ might not be equal to $B$. Because these two definitions are equivalent, then it must mean $A$ \ $B$ $\in E$ (if A is larger) or $B$ \ $A$ $\in E$ (if B is larger), right? – TSJ Feb 24 '15 at 10:40
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    $A=E\cup B$; this set is the closure of $E$, the smallest closed set containing $E$. Thus, $E$ is closed iff it already contains $B$, in which case of course it also contains $A$. – Brian M. Scott Feb 24 '15 at 10:47
  • Thanks Brian M. Scott. This makes it really clear what the relationship is between $A$, $B$, and $E$. Going off-topic a little - Rudin does not seem to talk about the set $A$ and its connection to limit points - is this something typically covered in more detail in a topology course or text? – TSJ Feb 24 '15 at 11:11
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    Yes. Between text proper and exercises, many topology texts will run through a fairly extensive catalogue of basic facts about such matters. – Brian M. Scott Feb 24 '15 at 11:26

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Since you used the tag "real analysis" I'm guessing you are working in the reals? As @Brian M. Scott mentioned your two definitions are equivalent in a metric space, so if you are working in the reals they are equivalent.

Hint to prove it: if $x$ is a limit point of $E$, consider $(x-1/n,x+1/n)\cap E$.

Brian McDonald
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