Question attached as an image, thank you! Prove the following: Suppose that {X; : i E I} and {Y; : j € J}are indexed families of topological spaces. Suppose that a : I → Jis a bijection and for all i E I, we have X; = Ya(1). Then II¡e1 X; ishomeomorphic to II;ej Yj.j€J 'j•

Name: Question attached as an image, thank you! Prove the following: Suppose
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Description: Question attached as an image, thank you! Prove the following: Suppose that {X; : i E I} and {Y; : j € J}are indexed families of topological spaces. Suppose that a : I → Jis a bijection and for all i E I, we have X; = Ya(1). Then II¡e1 X; ishomeomorp

let us define ϕ:∏i∈IXi→∏j∈JYj=∏i∈IXα(i) by ϕx1,x2,x3....=xα(1),xα2,..... For any x=x1,x2,.... and…

Answered: Prove the following: Suppose that {X; :…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question attached as an image, thank you!

Prove the following: Suppose that {X; : i E I} and {Y; : j € J}
are indexed families of topological spaces. Suppose that a : I → J
is a bijection and for all i E I, we have X; = Ya(1). Then II¡e1 X; is
homeomorphic to II;ej Yj.
j€J 'j•

Expert Solution

Step 1

let us define $ϕ : \prod_{i \in I} X_{i} \to \prod_{j \in J} Y_{j} = \prod_{i \in I} X_{α (i)}$

by $ϕ (x_{1}, x_{2}, x_{3} . . . .) = (x_{α (1)}, x_{α (2)}, . . . . .)$

For any $x = (x_{1}, x_{2}, . . . .) a n d y = (y_{1}, y_{2}, . . . . .)$
$ϕ (x) = ϕ (y) \Rightarrow x_{α (i)} = y_{α (i)} \forall i \in I \Rightarrow x_{j} = y_{j} \forall i \in J \Rightarrow x_{i} = y_{j} \forall i \in I a s I a n d J a r e b i j e c t i v e .$

$\Rightarrow \bar{x} = \bar{y}$

So, $ϕ$ is injective.

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