Exercise · ZW-PRODUCT-TOPOLOGY-008 · ★★☆ Intermediate
In \(\mathbb{R}\times\mathbb{R}\) with the product topology, which of these sets is NOT open?
Choose one option.
Show answer
- C. \(\{(x,y):\ y\le 0\}\)
Explanation
The half-plane \(y\le 0\) contains points of the line \(y=0\), and no open rectangle around them fits inside. The other three are open: \(x>0\) is \((0,\infty)\times\mathbb{R}\), and the other two are unions of open rectangles.
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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- ZW-PRODUCT-TOPOLOGY-005 Which statement about the open sets of the product topology on X× Y is correct?
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