- #1

mathmari

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MHB

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Let $\mathbb{R}$ provided with the metric $d(x,y)=|x-y|$. I want to check if the collections of sets $$S_1=\left \{\left (\frac{x}{2}, \frac{3x}{2}\right ): 0<x<1\right \}, \ \ \ \ \ S_2=\left \{\left (x-\frac{1}{2}, x+\frac{1}{2}\right ): 0<x<1\right \}$$ are open covers of $A=\left \{\frac{1}{n} : n\in \mathbb{N}\right \}\subset \mathbb{R}$.

An open cover is a collection of open sets whose union contains a given subset, right?

Could you give me a hint how we could check that in these cases? Do we have to check if the union of $S_1$ of all $x$ contains $A$ and the same also for $S_2$ ? (Wondering)