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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.

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.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.4: Spanning Sets And Linear Independence
Problem 26E: Determine whether the set S={2x+x2,8+x3,x2+x3,4+x2} spans P3.
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**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 \( X \), \(
Transcribed Image Text:**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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