Could you show me how to do 3.35 in detail? **Definition.** Let $X$ and $Y$ be two sets. The **projection functions** $\pi_X : X \times Y \rightarrow X$ and $\pi_Y : X \times Y \rightarrow Y$ are defined by $\pi_X(x,y) = x$ and $\pi_Y(x,y) = y$.**Definition.** Suppose $X$ and $Y$ are topological spaces. The **product topology** on the product $X \times Y$ is the topology whose basis is all sets of the form $U \times V$, where $U$ is an open set in $X$ and $V$ is an open set in $Y$.**Theorem 3.35.** Show that the product topology on $X \times Y$ is the same as the topology generated by the subbasis of inverse images of open sets under the projection functions, that is, the subbasis is $\{\pi_X^{-1}(U) \mid U \text{ open in } X\} \cup \{\pi_Y^{-1}(V) \mid V \text{ open in } Y\}$.

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Answered: U xV, where U is an

Math

Advanced Math

U xV, where U is an

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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Could you show me how to do 3.35 in detail?

**Definition.** Let $X$ and $Y$ be two sets. The **projection functions** $\pi_X : X \times Y \rightarrow X$ and $\pi_Y : X \times Y \rightarrow Y$ are defined by $\pi_X(x,y) = x$ and $\pi_Y(x,y) = y$.

**Definition.** Suppose $X$ and $Y$ are topological spaces. The **product topology** on the product $X \times Y$ is the topology whose basis is all sets of the form $U \times V$, where $U$ is an open set in $X$ and $V$ is an open set in $Y$.

**Theorem 3.35.** Show that the product topology on $X \times Y$ is the same as the topology generated by the subbasis of inverse images of open sets under the projection functions, that is, the subbasis is $\{\pi_X^{-1}(U) \mid U \text{ open in } X\} \cup \{\pi_Y^{-1}(V) \mid V \text{ open in } Y\}$.

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