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

In summary: This can be done by noting that if x_0 \in S \cap B_{\delta(x_0)}(x_0) then f^{-1}(O') \in B_{\delta(x_0)}(x_0) and so by the continuity of f we have f^{-1}(O') \in S \cap O'.
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This thread concerns an aspect of the proof of the theorem that states that a function is continuous on an open set if and only if the inverse image under f of every open set is open
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 4: Topology of R 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 ##f^{ -1 } (O') = S \cap O## for some open set ##O## ... ... "Can someone please explain why the above quoted statement is true ...

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My thoughts on this matter so far ...

Since ##f## is continuous at ##x_0## we can find ##\delta## such that

##f( S \cap B_\delta ( x_0 ) ) \subseteq B_\epsilon ( f(x_0) ) \subseteq O'##Now ... take inverse image under f of the above relationship (is this a legitimate move?)then we have ... ##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 ##B_\delta ( x_0 )## equal to ##O''## then we get##f^{ -1 } ( O' ) \supseteq S \cap O''## ...But now ... how do we find ##O## such that ##f^{ -1 } ( O' ) = S \cap O## ...?

Help will be appreciated ...

Peter
 
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  • #2
This could be wrong, so take the following hint with a grain of salt, because I don't have the time now to try it myself.

I think you can define ##O## to be the union of some open balls that the proof constructed.
 
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  • #3
Thanks Math_QED...

Reflecting on your suggestion...

Peter
 
  • #4
I think we are dealing with the restriction of [itex]f[/itex] to [itex]S[/itex], so [itex]f^{-1}(O') \subset S[/itex].
Now the [itex]\delta[/itex] constructed by the proof actually depends on [itex]x_0[/itex], so we can write [tex]
f^{-1}(O') \subseteq \bigcup_{x_0 \in f^{-1}(O')} S \cap B_{\delta(x_0)}(x_0) = S \cap \bigcup_{x_0 \in f^{-1}(O')} B_{\delta(x_0)}(x_0)[/tex] since [itex]x_0 \in S \cap B_{\delta(x_0)}(x_0)[/itex] and the equality follows from the fundamental laws of set algebra.

It remains to show inclusion in the opposite direction.
 
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1. What is the definition of continuity and open sets in mathematics?

Continuity is a mathematical concept that describes the smoothness or connectedness of a function. A function is continuous if it does not have any sudden jumps or breaks in its graph. Open sets, on the other hand, are subsets of a topological space that do not contain their boundary points. In simpler terms, they are sets that do not include their edge points.

2. How are continuity and open sets related?

Continuity and open sets are closely related concepts in mathematics. In fact, they are two of the fundamental concepts in topology, which is the study of the properties of spaces that are preserved under continuous transformations. In particular, a function is continuous if and only if the preimage of an open set is an open set.

3. What is the significance of Sohrab, Theorem 4.3.4 in the study of continuity and open sets?

Sohrab, Theorem 4.3.4 is a fundamental theorem in topology that states that a function is continuous if and only if the preimage of every open set is an open set. This theorem provides a necessary and sufficient condition for a function to be continuous, making it an essential tool in the study of continuity and open sets.

4. Can you give an example of a function that is continuous but does not have an open preimage?

Yes, a simple example is the function f(x) = 3x, which is continuous but does not have an open preimage. The preimage of any open set under this function would be an interval of the form (a/3, b/3), which is not an open set in the real numbers.

5. How are continuity and open sets used in real-world applications?

Continuity and open sets have numerous applications in real-world problems, particularly in physics, engineering, and economics. For example, in physics, continuity is used to describe the motion of objects, while open sets are used to represent the boundaries of physical systems. In engineering, these concepts are essential in the design and analysis of structures and systems. In economics, continuity and open sets are used to model and analyze economic systems and their behavior.

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