Inverse image of a set under the restriction of a function

[Math Amateur]

I am reading Munkres book, "Topology" (Second Edition).

I need help with an aspect of Theorem 18.2 Part (f) concerning the inverse image of a set under the restriction of a function ...

Theorem 18.2 Part f reads as follows:View attachment 4194
View attachment 4195
View attachment 4196In the above text we read:

" ... ... Let $$V$$ be an open set in $$Y$$.

Then

$$f^{-1} (V) \cap U_{ \alpha } = {(f | U_{ \alpha }) }^{-1} (V)$$ ... ...

... ... "I would like to prove that:

$$f^{-1} (V) \cap U_{ \alpha } = { (f | U_{ \alpha }) }^{-1} (V)
$$... BUT ... cannot see how to do this ...Can someone please help ...

Peter

 

[Fallen Angel]

Hi Peter,

I will just write down what this sets are.

$f^{-1}(V)=\{x\in X \ : \ f(x)\in V \}$

$f^{-1}(V)\cap U_{\alpha}=\{x\in U_{\alpha} \ : \ f(x)\in V \}$

Did you see now why the equality holds?

 

[Math Amateur]

Hi Peter,

I will just write down what this sets are.

$f^{-1}(V)=\{x\in X \ : \ f(x)\in V \}$

$f^{-1}(V)\cap U_{\alpha}=\{x\in U_{\alpha} \ : \ f(x)\in V \}$

Did you see now why the equality holds?

Thanks for the help, Fallen Angel ...