F is an open mapping implies f inverse cont.

Thread starter Unassuming
Start date Nov 21, 2008
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Inverse Mapping

In summary, an open mapping is a function that preserves the openness of sets, and its implication is that the function is continuous and the inverse image of an open set is also open. The open mapping theorem and the inverse function theorem are related, with the former being a generalization of the latter. A function can be an open mapping without being bijective, and the open mapping theorem is used in various real-world applications, particularly in engineering and science.

Nov 21, 2008

Unassuming

We have a continuous bijection f:X-->Y.

Prove that if f is open, then f inverse is continuous.

I can't figure it out.
"Proof". For V open in Y, there exists W open in X such that [tex]f[W] \subseteq V[/tex]. Where does the f is open definition apply?

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Nov 21, 2008

Unassuming

Ah! I got it. Posting always helps the blood get flowing. Trying to explain what I know to others.

1. What does it mean for a function to be an open mapping?

An open mapping is a function in which the image of an open set in the domain is also an open set in the range. In other words, the function preserves the openness of sets.

2. What is the implication of a function being an open mapping?

If a function is an open mapping, it means that the function is continuous and the inverse image of an open set is also an open set. This allows for easier analysis and manipulation of the function.

3. How does the open mapping theorem relate to the inverse function theorem?

The open mapping theorem and the inverse function theorem are closely related, as they both deal with the continuity and openness of functions. The open mapping theorem states that a continuous and open mapping has a continuous and open inverse, while the inverse function theorem states that a continuously differentiable function with a nonzero derivative has a continuously differentiable inverse. In essence, the open mapping theorem is a generalization of the inverse function theorem.

4. Can a function be an open mapping without being bijective?

Yes, a function can be an open mapping without being bijective. This is because the open mapping property only requires that the inverse image of an open set is open, not necessarily that the function is one-to-one and onto. An example of this is the exponential function, which maps open intervals to open intervals but is not bijective.

5. How is the open mapping theorem used in real-world applications?

The open mapping theorem is used in various real-world applications, particularly in engineering and science. It is used in the study of differential equations, optimization problems, and signal processing. It is also used in the development of algorithms and numerical methods for solving mathematical models and systems.