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- Jun 22, 2012

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I am focused on Chapter 4: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 4.25 ... ...

Theorem 4.25 (including its proof) reads as follows:

In the above proof by Apostol we read the following:

" ... ... The sets \(\displaystyle f^{ -1 } (A)\) form an open covering of \(\displaystyle X\) ... ... "

Could someone please demonstrate an explicit formal and rigorous proof of this statement ....?

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My thoughts:

Since \(\displaystyle f\) is continuous we have that each set \(\displaystyle f^{ -1 } (A)\) is open

and ...

... for \(\displaystyle X \subseteq S\) we have

\(\displaystyle X \subseteq f^{ -1 } ( f(x) )\) ... ... (see Apostol Exercise 2.7 (a) Chapter 2, page 44 ...)

... and we also have \(\displaystyle f(X) \subseteq A_c\) where \(\displaystyle A_c = \bigcup_{ A \in F } A\) ...

Therefore \(\displaystyle X \subseteq f^{ -1 } ( f(x) ) \subseteq f^{ -1 } ( A_c )\) ....

Is that correct? ... Does that constitute a formal and rigorous proof?

Hope someone can help ...

Peter

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The above post refers to Apostol Exercise 2.7 so I am providing access to the same as follows:

Hope that helps ...

Peter