Exercise · ZW-HAUSDORFF-SPACE-004 · ★☆☆ Basic
In \(X=\{1,2,3,4\}\) with open sets \(\varnothing, \{1\}, \{2\}, \{1,2\}, \{3,4\}, \{1,3,4\}, \{2,3,4\}, \{1,2,3,4\}\), how many pairs of distinct points can be separated by disjoint open sets?
Choose one option.
A 6 B 4 C 3 D 5 Show answer ← Back to the question
Explanation Only the pair \(\{3,4\}\) fails: every open set containing 3 or 4 contains both. The other 5 pairs are separated, for example by \(\{1\}\) and \(\{2\}\), or by \(\{1\}\) and \(\{3,4\}\).
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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