Exercise · ZW-TOPOLOGY-010 · ★★★ Advanced
Let \(X\) be infinite with open sets \(\varnothing\) and the sets with finite complement. If \(U\) and \(V\) are nonempty open sets, what happens?
Choose one option.
A \(U\cap V\) is empty if \(U\neq V\). B \(U\cup V=X\) always. C \(U\cap V\) is always nonempty. D \(U\cap V\) is finite. Show answer C. \(U\cap V\) is always nonempty.← Back to the question
Explanation The complements \(X\setminus U\) and \(X\setminus V\) are finite; so is their union, which cannot cover the infinite \(X\). A point outside both lies in \(U\cap V\).
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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