Exercise · ZW-HAUSDORFF-SPACE-002 · ★☆☆ Basic
How many of the 29 possible topologies on a 3-point set are Hausdorff?
Choose one option.
Show answer
- C. 1
Explanation
A finite Hausdorff space is discrete: each point is separated from the others by open sets, and the finite intersection of these gives \(\{x\}\) open. So only the discrete topology qualifies.
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 \(V\ni y\).
\[x\neq y\ \Rightarrow\ \exists\,U,V\ \text{open}:\ x\in U,\ y\in V,\ U\cap V=\varnothing\]
More practice problems: hausdorff space
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- ZW-HAUSDORFF-SPACE-008 Let f:X→ Y be continuous and injective, with Y Hausdorff. What follows about X?
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