Exercise · ZW-PRODUCT-TOPOLOGY-005 · ★☆☆ Basic
Which statement about the open sets of the product topology on \(X\times Y\) is correct?
Choose one option.
Show answer
- A. Every open set is a union of sets \(U\times V\) with \(U,V\) open
Explanation
The products \(U\times V\) form a basis, and open sets are unions of basis elements. Not every open set is a product (some unions are not), but every union of products is open.
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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