**Theorem:**Let $\{X_i : i \in I\}$ be an indexed family of topological spaces. Suppose that $I$ is the union of two disjoint subsets $J$ and $K$. Then\[\prod_{i \in I} X_i\]is homeomorphic to\[\prod_{i \in J} X_i \times \prod_{i \in K} X_i\]**Explanation of Symbols:**- $\prod_{i \in I} X_i$ denotes the product topology of the family of spaces indexed by $I$.- The symbol $\times$ represents the Cartesian product of two topological spaces.- The condition $I = J \cup K$ indicates that the index set $I$ is split into two disjoint subsets $J$ and $K$.**Concepts:**- **Homeomorphism:** A bijective continuous function with a continuous inverse between two topological spaces, indicating that the spaces are topologically equivalent.- **Product Topology:** The topology on the Cartesian product of a collection of topological spaces, which is the coarsest topology for which all the projections are continuous.This assertion holds within the field of topology, illustrating a fundamental property of product spaces and their decomposition into subproducts.

To show a homeomorphism between two sets, we need to find a function f: ∏i∈IXi→∏i∈JXi×∏i∈KXi, such…

Answered: Prove the following: Let {X; : i e I}…

Prove the following: Let {X; : i e I} be an indexed family of topo- logical spaces. Suppose that I is the union of two disjoint subsets J and K. Then II X: ieI is homeomorphic to II X; x IIX: ieJ iEK

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

**Theorem:**

Let $\{X_i : i \in I\}$ be an indexed family of topological spaces. Suppose that $I$ is the union of two disjoint subsets $J$ and $K$. Then

\[
\prod_{i \in I} X_i
\]

is homeomorphic to

\[
\prod_{i \in J} X_i \times \prod_{i \in K} X_i
\]

**Explanation of Symbols:**

- $\prod_{i \in I} X_i$ denotes the product topology of the family of spaces indexed by $I$.
- The symbol $\times$ represents the Cartesian product of two topological spaces.
- The condition $I = J \cup K$ indicates that the index set $I$ is split into two disjoint subsets $J$ and $K$.

**Concepts:**

- **Homeomorphism:** A bijective continuous function with a continuous inverse between two topological spaces, indicating that the spaces are topologically equivalent.
- **Product Topology:** The topology on the Cartesian product of a collection of topological spaces, which is the coarsest topology for which all the projections are continuous.

This assertion holds within the field of topology, illustrating a fundamental property of product spaces and their decomposition into subproducts.

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