Exercise · ZW-HAUSDORFF-SPACE-001 · ★☆☆ Basic
Which of these topologies on \(X=\{1,2,3\}\) is Hausdorff?
Choose one option.
Show answer
- B. \(\varnothing, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\)
Explanation
Only the discrete topology (all subsets) separates every pair: \(\{x\}\) and \(\{y\}\) are disjoint open sets. In the others some pair of points always has overlapping neighbourhoods.
Review the concept: Hausdorff space
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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