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What topological property does $\Bbb R^n$ have which accounts for only the trivial sets ($\emptyset, \Bbb R^n$) being clopen? Is there a more general type of space where this is true? thanks

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The basic way to prove this is to first prove it is true for $[0,1]$ then show that any path-connected space necessarily has this property. So, while this is, by definition, the concept of a "connected" topological space, the underlying intuition for why it is true is that $\mathbb R^n$ is path-connected. – Thomas Andrews Feb 5 '13 at 21:23
Also, you have to be sure you're using the usual topology on $\Bbb R^n$. In the indiscrete topology, every set is connected, while in the discrete topology, only singletons are connected. – Clayton Feb 5 '13 at 21:25
That works, @JonasMeyer, but it isn't the most basic (and doesn't generalize well - for example, path connectedness gets you the punctured Euclidean space pretty easily.) Basically, path-connectedness is our early intuition about what "connected" means, and, although they are not equivalent, it is the most intuitive way to show connectedness. – Thomas Andrews Feb 5 '13 at 21:57

By definition, this "general type" of space is connected spaces.

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