Exercise · ZW-TOPOLOGY-003 · ★☆☆ Basic
Each interval \((-1/n,1/n)\) is open in \(\mathbb{R}\). What is the intersection of all of them, \(n\ge1\)?
Choose one option.
A \((-1,1)\), which is open. B \(\varnothing\), which is open. C \(\{0\}\), which is open. D \(\{0\}\), which is not open: an infinite intersection of open sets need not be open. Show answer D. \(\{0\}\), which is not open: an infinite intersection of open sets need not be open.← Back to the question
Explanation A number \(x\ne0\) lies outside \((-1/n,1/n)\) for large \(n\), so only \(0\) belongs to all of them. No interval around \(0\) fits inside \(\{0\}\); that is why the definition asks only for finite intersections.
Review the concept: Topology A topology on a set \(X\) is a family \(\mathcal{T}\) of subsets of \(X\), whose members \(U\) are called open sets , containing \(\varnothing\) and \(X\) and closed under arbitrary unions and finite intersections.
\[\varnothing,X\in\mathcal{T};\quad U_i\in\mathcal{T}\Rightarrow\bigcup_i U_i\in\mathcal{T};\quad U_1\cap U_2\in\mathcal{T}\]
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