Exercise · ZW-PRODUCT-TOPOLOGY-004 · ★☆☆ Basic
With \(S=\{0,1\}\) with open sets \(\varnothing,\{1\},S\) and \(T=\{a,b,c\}\) with open sets \(\varnothing,\{a\},\{a,b\},T\) and the projection \(\pi_1:S\times T\to S\), how many points does \(\pi_1^{-1}(\{1\})\) have?
Choose one option.
A \(1\) B \(3\) C \(2\) D \(6\) Show answer ← Back to the question
Explanation \(\pi_1^{-1}(\{1\})=\{1\}\times T\) has 3 points and is open because \(\{1\}\) and \(T\) are; this is why \(\pi_1\) is continuous.
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-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-007 Which is a basis for the usual (product) topology on ℝ×ℝ?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…