How do I show 7.35? Could you explain this in great detail? Thank you! **Definition**: Suppose $ \mathcal{T} $ and $ \mathcal{T}' $ are two topologies on the same underlying set $ X $. If $ \mathcal{T} \subset \mathcal{T}' $, then we say $ \mathcal{T}' $ is finer than $ \mathcal{T} $. Alternatively, we say $ \mathcal{T} $ is coarser than $ \mathcal{T}' $. We say strictly coarser or strictly finer if additionally $ \mathcal{T} \neq \mathcal{T}' $.**Definition**: A function $ f : X \to Y $ is an *embedding* if and only if $ f : X \to f(X) $ is a homeomorphism from $ X $ to $ f(X) $, where $ f(X) $ has the subspace topology from $ Y $.**Definition**: The *projection maps* $ \pi_X : X \times Y \to X $ and $ \pi_Y : X \times Y \to Y $ are defined by $ \pi_X(x, y) = x $ and $ \pi_Y(x, y) = y $.**Theorem 7.32**: Let $ X $ and $ Y $ be topological spaces. The projection maps $ \pi_X, \pi_Y $ on $ X \times Y $ are continuous, surjective, and open.In fact, the topology on the product space can be characterized as the coarsest topology that makes the projection maps continuous.**Theorem 7.33**: Let $ X $ and $ Y $ be topological spaces. The product topology on $ X \times Y $ is the coarsest topology on $ X \times Y $ that makes the projection maps $ \pi_X, \pi_Y $ on $ X \times Y $ continuous.**Theorem 7.35**: Let $ X $ and $ Y $ be topological spaces. For every $ y \in Y $, the subspace $ X \times \{ y \} $ of $ X \times Y $ is homeomorphic to $ X $.**Theorem 7.36**: Let $ X $, \(

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Description: How do I show 7.35? Could you explain this in great detail? Thank you! **Definition**: Suppose $ \mathcal{T} $ and $ \mathcal{T}' $ are two topologies on the same underlying set $ X $. If $ \mathcal{T} \subset \mathcal{T}' $, then we say \( \

Answered: Image /qna-images/answer/7ba14734-44f0-4622-9cf8-76d872f334dc.jpg

Answered: Theorem 7.35. Let X and Y be…

Math

Advanced Math

Theorem 7.35. Let X and Y be topological spaces. For every y E Y, the subspace X × {y} of X × Y is homeomorphic to X.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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How do I show 7.35? Could you explain this in great detail? Thank you!

**Definition**: Suppose $ \mathcal{T} $ and $ \mathcal{T}' $ are two topologies on the same underlying set $ X $. If $ \mathcal{T} \subset \mathcal{T}' $, then we say $ \mathcal{T}' $ is finer than $ \mathcal{T} $. Alternatively, we say $ \mathcal{T} $ is coarser than $ \mathcal{T}' $. We say strictly coarser or strictly finer if additionally $ \mathcal{T} \neq \mathcal{T}' $.

**Definition**: A function $ f : X \to Y $ is an *embedding* if and only if $ f : X \to f(X) $ is a homeomorphism from $ X $ to $ f(X) $, where $ f(X) $ has the subspace topology from $ Y $.

**Definition**: The *projection maps* $ \pi_X : X \times Y \to X $ and $ \pi_Y : X \times Y \to Y $ are defined by $ \pi_X(x, y) = x $ and $ \pi_Y(x, y) = y $.

**Theorem 7.32**: Let $ X $ and $ Y $ be topological spaces. The projection maps $ \pi_X, \pi_Y $ on $ X \times Y $ are continuous, surjective, and open.

In fact, the topology on the product space can be characterized as the coarsest topology that makes the projection maps continuous.

**Theorem 7.33**: Let $ X $ and $ Y $ be topological spaces. The product topology on $ X \times Y $ is the coarsest topology on $ X \times Y $ that makes the projection maps $ \pi_X, \pi_Y $ on $ X \times Y $ continuous.

**Theorem 7.35**: Let $ X $ and $ Y $ be topological spaces. For every $ y \in Y $, the subspace $ X \times \{ y \} $ of $ X \times Y $ is homeomorphic to $ X $.

**Theorem 7.36**: Let $ X $, \(

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