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I learned in different math courses, that $\mathbb{C}$ is a clopen set. I'm very uncertain if I understand this right, please tell me if these two examples are true:

  • $\mathbb{C} \setminus \{ 0 \}$ is open,
  • $\mathbb{C} \setminus \{ z \in \mathbb{C} : |z| < 1 \}$ is closed.

Thanks.

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    $\begingroup$ The examples are true. But they don't show that $\mathbb{C}$ is clopen. $\endgroup$ – Raskolnikov Jan 11 '11 at 14:28
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A clopen set is a set which is both closed and open.

When talking about a space, it is always clopen in itself. This is because we want both the space and the empty set to be open.

So $\mathbb{C}$ is indeed clopen in $\mathbb{C}$ (in any given topology). However, note that $\mathbb{R}$ is clopen in $\mathbb{R}$, but as a subset of $\mathbb{C}$ it's actually just closed. So when you move from $\mathbb{C}$ to something bigger (for example the Riemann sphere) then $\mathbb{C}$ is open, and not closed.

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  • $\begingroup$ +1 for the hint, that $\mathbb{R} \subset \mathbb{C}$ is closed, but clopen for itself. Thanks. $\endgroup$ – a1337q Jan 11 '11 at 14:52
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    $\begingroup$ And in a connected space the only clopen sets are the empty set and the whole space. $\endgroup$ – Jonas Teuwen Jan 11 '11 at 15:00
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    $\begingroup$ And when you move from C to C^2, then copies of C are closed, and not open. The point is that "open" and "closed" are always relative notions and should never be used absolutely. $\endgroup$ – Qiaochu Yuan Jan 11 '11 at 16:44
  • $\begingroup$ To build on Qiaochu's comment, some have suggested that compact (Hausdorff) is the appropriate "absolute" version of closed. One shouldn't take this too literally, but it can be a helpful heuristic. $\endgroup$ – Pete L. Clark Jan 11 '11 at 19:16
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The whole space is always clopen in any topology. That is part of the definition of a topology. Your two examples are correct.

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    $\begingroup$ The choice of words is a bit unfortunate. Perhaps it is better to use "whole" rather than "complete". $\endgroup$ – Andrés E. Caicedo Jan 11 '11 at 18:28
  • $\begingroup$ @Andres: I agree. Thanks $\endgroup$ – Ross Millikan Jan 11 '11 at 18:41
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$\mathbb{C}$ is indeed clopen since it is both open and closed. Note that one way to define a connected space in topology is that there are no non-trivial clopen sets (the empty set and the whole set will trivially be clopen regardless of the topology).

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