Analysis: Continuous open mappings.

In summary, the conversation discusses the concept of open mappings and their properties, specifically in regards to continuous open mappings of the Reals into the Reals. The question of whether or not all continuous open mappings are also monotonic is raised, using the example of the function f(x)= x^{2}. It is ultimately concluded that f(x)= x^{2} is not a continuous open mapping of the Reals into the Reals that is not monotonic.
  • #1
futurebird
272
0
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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  • #2
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))?
 
  • #3
morphism said:
What is f((-1,1))?

[0, 1)

THANKS. Got it now.
 

FAQ: Analysis: Continuous open mappings.

What is the definition of a continuous open mapping?

A continuous open mapping is a function between two topological spaces, where the preimage of any open set in the range is an open set in the domain.

How is a continuous open mapping different from a continuous mapping?

A continuous mapping only requires that the preimage of open sets in the range are open or closed in the domain, while a continuous open mapping specifically requires them to be open.

What is the importance of continuous open mappings in analysis?

Continuous open mappings are important in analysis because they preserve important properties such as connectedness and compactness, making them useful tools in proving theorems and solving problems.

Can a continuous open mapping be bijective?

Yes, a continuous open mapping can be bijective. However, it is not always the case and it depends on the specific mapping and the topological spaces involved.

How do continuous open mappings relate to the concept of homeomorphism?

A continuous open mapping is a special case of a homeomorphism, where the mapping is not only continuous and open, but also has an inverse that is also continuous and open. Homeomorphisms are important in understanding the topological structure of spaces and continuous open mappings are a specific type of homeomorphism.

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