How do I show 7.33? 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.

Name: How do I show 7.33? Could you explain this in great detail? Thank you!
Uploaded: 2024-03-20T07:06:16.000Z
Channel: Bartleby
Description: How do I show 7.33? 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 \( \