Exercise · ZW-HAUSDORFF-SPACE-007 · ★★☆ Intermediate
In \(X=\{1,2,3,4\}\) with open sets \(\varnothing, \{1\}, \{4\}, \{1,4\}, \{2,3\}, \{1,2,3\}, \{2,3,4\}, \{1,2,3,4\}\), which pair of points cannot be separated by disjoint open sets?
Choose one option.
A \(\{1,2\}\) B \(\{2,3\}\) C \(\{1,4\}\) D \(\{3,4\}\) Show answer ← Back to the question
Explanation Every open set containing 2 contains 3 and vice versa, since \(\{2,3\}\) is indivisible. The other pairs are separated by \(\{1\}\), \(\{4\}\) or \(\{2,3\}\).
A topological space \(X\) is Hausdorff if for every pair of distinct points \(x\) and \(y\) there exist disjoint open sets \(U\ni x\) and \(V\ni y\).
\[x\neq y\ \Rightarrow\ \exists\,U,V\ \text{open}:\ x\in U,\ y\in V,\ U\cap V=\varnothing\]
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