Theorem 3.37. A basis for the product topology on [Icej Xa is the collection of all sets of the form IIce Ua, where Ug is open in Xa for each a and Ug = Xa for all but finitely тапy a.
Theorem 3.37. A basis for the product topology on [Icej Xa is the collection of all sets of the form IIce Ua, where Ug is open in Xa for each a and Ug = Xa for all but finitely тапy a.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Could you show me how to do 3.37 in detail?
![**Definition.** Let \(\{X_{\alpha}\}_{\alpha \in \lambda}\) be a collection of topological spaces. The product \(\prod_{\alpha \in \lambda} X_{\alpha}\), or Cartesian product, is the set of functions
\[
\{f : \lambda \rightarrow \bigcup_{\alpha \in \lambda} X_{\alpha} \mid \text{for all } \alpha \in \lambda, f(\alpha) \in X_{\alpha}\}.
\]
Here \(f(\alpha)\) is called the \(\alpha\)th coordinate of \(f\). The spaces \(X_{\alpha}\) are sometimes called factors of the infinite product. Thus a point in the product may be thought of as a function that associates to each \(\alpha\) an element \(f(\alpha)\) of the factor \(X_{\alpha}\).
**Definition.** For each \(\beta\) in \(\lambda\), define the projection function \(\pi_{\beta} : \prod_{\alpha \in \lambda} X_{\alpha} \rightarrow X_{\beta}\) by \(\pi_{\beta}(f) = f(\beta)\). We define the product topology on \(\prod_{\alpha \in \lambda} X_{\alpha}\) to be the one generated by the subbasis of sets of the form \(\pi_{\beta}^{-1}(U_{\beta})\), where \(U_{\beta}\) is open in \(X_{\beta}\).
**Theorem 3.37.** A basis for the product topology on \(\prod_{\alpha \in \lambda} X_{\alpha}\) is the collection of all sets of the form \(\prod_{\alpha \in \lambda} U_{\alpha}\), where \(U_{\alpha}\) is open in \(X_{\alpha}\) for each \(\alpha\) and \(U_{\alpha} = X_{\alpha}\) for all but finitely many \(\alpha\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbfa77723-e387-48dc-857e-67a9efe99fb1%2Ffc5da36d-f5fd-4553-bac3-d3b6fbe70228%2Fr7lg6xn_processed.png&w=3840&q=75)
Transcribed Image Text:**Definition.** Let \(\{X_{\alpha}\}_{\alpha \in \lambda}\) be a collection of topological spaces. The product \(\prod_{\alpha \in \lambda} X_{\alpha}\), or Cartesian product, is the set of functions
\[
\{f : \lambda \rightarrow \bigcup_{\alpha \in \lambda} X_{\alpha} \mid \text{for all } \alpha \in \lambda, f(\alpha) \in X_{\alpha}\}.
\]
Here \(f(\alpha)\) is called the \(\alpha\)th coordinate of \(f\). The spaces \(X_{\alpha}\) are sometimes called factors of the infinite product. Thus a point in the product may be thought of as a function that associates to each \(\alpha\) an element \(f(\alpha)\) of the factor \(X_{\alpha}\).
**Definition.** For each \(\beta\) in \(\lambda\), define the projection function \(\pi_{\beta} : \prod_{\alpha \in \lambda} X_{\alpha} \rightarrow X_{\beta}\) by \(\pi_{\beta}(f) = f(\beta)\). We define the product topology on \(\prod_{\alpha \in \lambda} X_{\alpha}\) to be the one generated by the subbasis of sets of the form \(\pi_{\beta}^{-1}(U_{\beta})\), where \(U_{\beta}\) is open in \(X_{\beta}\).
**Theorem 3.37.** A basis for the product topology on \(\prod_{\alpha \in \lambda} X_{\alpha}\) is the collection of all sets of the form \(\prod_{\alpha \in \lambda} U_{\alpha}\), where \(U_{\alpha}\) is open in \(X_{\alpha}\) for each \(\alpha\) and \(U_{\alpha} = X_{\alpha}\) for all but finitely many \(\alpha\).
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