Exercise · ZW-PRODUCT-TOPOLOGY-007 · ★★☆ Intermediate
Which is a basis for the usual (product) topology on \(\mathbb{R}\times\mathbb{R}\)?
Choose one option.
Show answer
- B. The open rectangles \((a,b)\times(c,d)\)
Explanation
Open intervals form a basis of the real line, so products of two of them, the open rectangles, form a basis of the plane. Closed rectangles are not open, and neither are vertical lines.
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
- ZW-PRODUCT-TOPOLOGY-001 Let S=\0,1\ with open sets ∅,\1\,S and T=\a,b,c\ with open sets ∅,\a\,\a,b\,T. How many distinct se…
- ZW-PRODUCT-TOPOLOGY-002 With S=\0,1\ with open sets ∅,\1\,S and T=\a,b,c\ with open sets ∅,\a\,\a,b\,T, how many open sets…
- ZW-PRODUCT-TOPOLOGY-003 With S=\0,1\ with open sets ∅,\1\,S and T=\a,b,c\ with open sets ∅,\a\,\a,b\,T, which of these subs…
- ZW-PRODUCT-TOPOLOGY-004 With S=\0,1\ with open sets ∅,\1\,S and T=\a,b,c\ with open sets ∅,\a\,\a,b\,T and the projection π…
- ZW-PRODUCT-TOPOLOGY-005 Which statement about the open sets of the product topology on X× Y is correct?
- ZW-PRODUCT-TOPOLOGY-006 With S=\0,1\ with open sets ∅,\1\,S and T=\a,b,c\ with open sets ∅,\a\,\a,b\,T, how many points doe…
- ZW-PRODUCT-TOPOLOGY-008 In ℝ×ℝ with the product topology, which of these sets is NOT open?
- ZW-PRODUCT-TOPOLOGY-009 With S=\0,1\ with open sets ∅,\1\,S and T=\a,b,c\ with open sets ∅,\a\,\a,b\,T, how many open sets…