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I realise this is a very easy question. But it seems to me that from the standard (open sets) definition of a topology ($X$ and $\varnothing$ open, closed under arbitrary unions and finite intersections) that the collection $\{\Bbb R, [0, 1], \varnothing\}$ forms a topology on $\mathbb R$. Why is this not the case?

Also, can someone point me in the direction of a good proof for why an open set in a topology does not contain it's boundary points (a proof from the axioms of a topology not from the concept of an open set in a metric space).

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    Thanks to everyone for the help2012-09-23

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$\{\mathbb R,[0,1],\varnothing\}$ is a topology on $\mathbb R$. Also, an open set may contain its boundary; for example, $\mathbb R$ itself. This uses the definition that the boundary of a set is the intersection of its closure and the closure of its complement, so that the boundary of $\mathbb R$ is $\varnothing$.

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    Fair enough, I'll delete all these comments in a minute. You may want to include a note about how this is the *only* way an open set can contain its boundary: If the boundary is nonempty, an open set *never* contains any of the boundary points.2012-09-23