Exercise · ZW-PRODUCT-TOPOLOGY-010 · ★★★ Advanced
Let \(S=\{0,1\}\) with open sets \(\varnothing,\{1\},S\). How many open sets does the product topology on \(S\times S\) have?
Choose one option.
Show answer
- C. \(6\)
Explanation
The basic products are \(\varnothing\) and the \(2\times 2=4\) nonempty products; their unions give 6 distinct open sets.
Review the concept: Product topology
The topology on the product \(X\times Y\) of two spaces \((X,\mathcal{T}_X)\) and \((Y,\mathcal{T}_Y)\) that has as basis \(\mathcal{B}\) the sets \(U\times V\) with \(U\) and \(V\) open.
\[\mathcal{B}=\{U\times V:\ U\in\mathcal{T}_X,\ V\in\mathcal{T}_Y\}\]
More practice problems: product topology
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