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The idea is to prove that the open intervals (like $]a,b[$) are contained in the topology of the left-closed sets ($[a,b[$), but I cannot see a way of generating open sets from half-closed sets. (Same goes for the right-closed sets)

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2 Answers 2

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$ ]a,b[ = \bigcup_{n\in\mathbb{N}} [a + \frac{1}{n},b[$

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Hint: Note that the sequence $\frac1n\to 0$, but $0\notin\{\frac1n:n\in\mathbb N\}$.

Can you think of a way to use this with a family of intervals to prove what you want?

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