2
$\begingroup$

Let $X$ be a metric space. Furthermore, let $E$ be an open subset of $X$. Then, the complement of $E$, or all members of $X$ that are not in $E$, is closed, or contains all of its limit points. I understand this to be true locally around $E$.

However, why is this true when taking into account $X$ entirely. For instance, could there not exist a limit point of $X$ which is not a limit point of $E$? What if there is a point not in $E$ "distant" from $E$ which is a limit point of $X$ but not in $X$? Then, $E$ would still be open, but its complement would not be closed.

  • 1
    In order to be closed, the complement $E^c$ needs to contain all limits of convergent sequences *in* $E^c$, not all limits of convergent sequences *in* $E$. Does that help? (Your remarks about points outside of $X$ puzzle me: openness and closedness are *relative* to $X$; even if there are any points outside of $X$ (there need not be in your setup), then you should ignore them in this context.)2012-01-01
  • 0
    "a limit point of $X$ but not in $X$" - if $X$ is the metric space you are working in, it does not make sense to talk about things "outside" of $X$, $X$ is all you have.2012-01-01
  • 0
    @Santiago: I think the question should be rephrased to take what you point out into account!2012-01-01

2 Answers 2