Exercise · ZW-HAUSDORFF-SPACE-009 · ★★★ Advanced

Let \(X=\{a,b\}\) have the trivial topology and \(Y\) any Hausdorff space with two or more points. Is \(Y\times X\) with the product topology Hausdorff?

Choose one option.

Show answer
  • A. No: \((y,a)\) and \((y,b)\) cannot be separated

Explanation

Every open set of \(Y\times X\) containing \((y,a)\) also contains \((y,b)\), because the open sets of \(X\) are \(\varnothing\) and \(X\). A product is Hausdorff only if every factor is.

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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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