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So, a subset, $U$, of $\mathbb{R}^2$ is open if we can pick any point, $x \in U$ such that for some real number $\epsilon > 0$, given any point $y \in \mathbb{R}^2$ a distance less than $\epsilon$ from $x$, then $y \in U$.

Well, in $\mathbb{R}^2$ our neighborhoods are then disks, but a line, say $y = 5$ does not contain any points above or below 5, but this can't be possible in an open set because our open set would have to contain some points due it being a disk. The same could be said about points, but with points we're not talking simply about above or below, we're discussing every direction.

My explanation may not be mathematically rigorous, but I'm just asking if my intuition is correct.

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  • $\begingroup$ Yes, good intuition. $\endgroup$ – Ian Coley Apr 3 '14 at 6:33
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    $\begingroup$ Keep in mind that closed is very different from not open. $\endgroup$ – Alex Becker Apr 3 '14 at 6:36
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    $\begingroup$ Well, there are plenty of sets that are neither open nor closed. This is good intuition for knowing that these sets aren't open, but not that they're closed. Intuition for closed sets should either come from thinking about their complements, or thinking about limits of points in these sets. $\endgroup$ – user98602 Apr 3 '14 at 6:36
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    $\begingroup$ Ah yes, another way to think about it is that their complements are open. After all, the definition in my textbook about closed sets is that they're the complements of open sets. $\endgroup$ – David Apr 3 '14 at 6:38
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    $\begingroup$ @David It is not another way to think about it. It is the way to think about it. All you explained in your question is why lines and points are not open. For example $[0,1)\subseteq\mathbb R$ is neither open nor closed! $\endgroup$ – Christoph Apr 3 '14 at 6:49
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You have the right idea. These sets are closed because their complements are open. You can see that because if you pick any point in the complement of such a set, you can form an open ball about that point that misses the set (take the radius to be the distance from the point to the set, or, if you want to be even more sure, half that).

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