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$\overline{\mathbb{R}}$ is a topological space, but not a metric space, so I'm not sure if it is true.

Let $U$ be an open set in $\overline{\mathbb{R}}$ such that $\infty\in U$.

Then, how do I that prove there exists $c\in\mathbb{R}$ such that $(c,\infty] \subset U$ and such that $(c,\infty]$ is an open set?

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    Yes, your characterisation of the open subsets of $\overline{\mathbb{R}}$ is correct. (Or, at least, this is the natural way of defining the topology on this set.)2012-12-07

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