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Continuity and Open Sets ... Sohrab, Theorem 4.3.4 ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 4: Topology of [FONT=MathJax_AMS]R[/FONT] and Continuity ... ...

I need help in order to fully understand the proof of Theorem 4.3.4 ... ...


Theorem 4.3.4 and its proof read as follows:




Sohrab - Theorem 4.3.4 ... .png




In the above proof by Sohrab we read the following:

" ... ... Therefore \(\displaystyle f^{ -1 } (O') = S \cap O\) for some open set \(\displaystyle O\) ... ... "


Can someone please explain why the above quoted statement is true ...


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***EDIT *** ... ... My thoughts on this matter so far ...

Since \(\displaystyle f\) is continuous at \(\displaystyle x_0\) we can find \(\displaystyle \delta\) such that

\(\displaystyle f( S \cap B_\delta ( x_0 ) ) \subseteq B_\epsilon ( f(x_0) ) \subseteq O'\)


Now ... take inverse image under \(\displaystyle f\) of the above relationship (is this a legitimate move?)


then we have ...


\(\displaystyle S \cap B_\delta ( x_0 ) \subseteq f^{ -1 } ( B_\epsilon ( f(x_0) ) ) \subseteq f^{ -1 } ( O' )\)


So that ... if we put the open set \(\displaystyle B_\delta ( x_0 )\) equal to \(\displaystyle O''\) then we get


\(\displaystyle f^{ -1 } ( O' ) \supseteq S \cap O''\) ...


But now ... how do we find \(\displaystyle O\) such that


\(\displaystyle f^{ -1 } ( O' ) = S \cap O\) ...

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Help will be appreciated ...

Peter
 
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