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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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    What makes you think it's not a topology?2012-09-23
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    After your most recent revision of the question, what you have is a topology.2012-09-23
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    That's what I thought, I think I was just thrown by the examples of topologies on R induced by metrics.2012-09-23
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    Thanks to everyone for the help2012-09-23

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