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My book says that a metric space $E$ is connected if the only subsets of $E$ that are both open and closed are $E$ and $\varnothing$.

It goes on to say that if a space $E$ is not connected then it has a subset $A$ that is both open and closed and whose complement is also both open and closed.

Then it says that $E$ is the union of these two sets, both of which are open … .

So how did $A$ and its complement go from being open and closed to open? (I don't really know what it means to be both open and closed either; only closed, open or neither).

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A set being both open and closed simply means that it has the property that it is open, and the property that it is closed. That is, both it and its complement are open.

For example, in $\mathbb{R}$, the set $\mathbb{R}$ is open. However, $\mathbb{R}$ is also closed in $\mathbb{R}$, since its complement in $\mathbb{R}$, $\emptyset$, is open. Similarly, $\emptyset$ is both open and closed in $\mathbb{R}$, as it and its complement are both open.

It is fairly easy to see that in any space $X$, both $\emptyset$ and $X$ must be both open and closed, using the same sort of reasoning as with $\mathbb{R}$ above.

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  • $\begingroup$ Oh okay that makes sense now thx. $\endgroup$ – rummi Aug 5 '13 at 22:19

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