In trying to solve a complex problem, I encountered a sub problem about homeomorphisms between level sets of functions. To explain the problem I have created a specific example pictured in a sketch below:
Additional information about the example:
- The black lines are level sets of some function which is defined for some space $\mathbb{R}^n$ (for convenience the sketch is for $\mathbb{R}^2$). The red line is to indicate there is a discontinuity between the level sets between the two regions $V_{1,1}$ and $V_{2,1}$ (and the other two regions $V_{1,2}$ and $V_{2,2}$).
- Points on the red boundary belong to region $V_{2,1}$ (and $V_{2,2}$).
- There exists a homeomorphism between the level sets for regions $V_{1,1}$ and $V_{1,2}$, and a homeomorphism between the level sets for regions $V_{2,1}$ and $V_{2,2}$.
My question is: With this discontinuity between $V_{1,1}$ and $V_{2,1}$ (or the other two regions $V_{1,2}$ and $V_{2,2}$) can there exist a homeomorphism between the two different level sets.
My concern is at the red boundary line and the open set definition of continuity, as I do not believe that open sets maps to open sets along this red boundary. On the other hand the existence of homeomorphisms for the different regions suggests that I could combine them to form a single homeomorphism.
Additional Notes
- If you have any additional questions I can provide needed clarification.
Best Answer
Yes: in particular, if $V_{1,1} = V_{1,2}$ and $V_{2,1} = V_{2,2}$, then the identity is such a map, and is always a homeomorphism, without any assumptions on the nature of the space involved.