Analysis: Continuous open mappings.

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SUMMARY

The discussion centers on the properties of continuous open mappings from the Reals to the Reals, specifically addressing the claim that every continuous open mapping is monotonic. The user questions the validity of this assertion by presenting the function f(x) = x², which is continuous but not monotonic, and challenges its classification as a continuous open mapping. The conclusion drawn is that f(x) = x² does not map open sets to open sets, as evidenced by the mapping of the interval (-1, 1) to [0, 1), which is not open in the Reals.

PREREQUISITES
  • Understanding of continuous functions in topology
  • Familiarity with open sets in metric spaces
  • Knowledge of monotonic functions and their properties
  • Basic concepts of real analysis, particularly mappings
NEXT STEPS
  • Study the definition and properties of continuous functions in topology
  • Explore the concept of open sets and their significance in real analysis
  • Investigate monotonic functions and their implications in mappings
  • Examine counterexamples in real analysis to solidify understanding of continuous open mappings
USEFUL FOR

Mathematicians, students of real analysis, and anyone interested in the properties of continuous functions and their mappings in topology.

futurebird
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Here is a mystifying question from Rudin Chapter 4, #15

Call a mapping of X into Y "open" if f(V) is an open set in Y whenever V is an open set in X. Prove that every continuous open mapping of Reals into the Reals is monotonic.​

I'm having trouble proving this, in part, because I don't even think it's true. Wouldn't say... [tex]f(x)= x^{2}[/tex] map open sets to open sets? And [tex]f(x)= x^{2}[/tex] isn't monotonic on the Reals. Can someone tell me why [tex]f(x)= x^{2}[/tex] isn't a continuous open mapping of Reals into the Reals that is NOT monotonic.
 
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futurebird said:
I'm having trouble proving this, in part, because I don't even think it's true. Wouldn't say... [tex]f(x)= x^{2}[/tex] map open sets to open sets? And [tex]f(x)= x^{2}[/tex] isn't monotonic on the Reals. Can someone tell me why [tex]f(x)= x^{2}[/tex] isn't a continuous open mapping of Reals into the Reals that is NOT monotonic.
What is f((-1,1))?
 
morphism said:
What is f((-1,1))?

[0, 1)

THANKS. Got it now.
 

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