Convexity and Topology: Are They Related?

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SUMMARY

The discussion centers on the relationship between convexity and topology, specifically whether two homeomorphic sets can guarantee that if one is convex, the other must also be convex. The consensus is that this is not necessarily true, as homeomorphism preserves topological properties but does not imply convexity. Therefore, a convex set does not ensure that its homeomorphic counterpart retains the same convex characteristics.

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  • Understanding of basic topology concepts, including homeomorphism.
  • Familiarity with convex sets and their properties.
  • Knowledge of mathematical proofs and logical reasoning.
  • Basic grasp of set theory and its applications in topology.
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  • Research the definition and properties of homeomorphic sets in topology.
  • Study the characteristics of convex sets and their implications in various mathematical contexts.
  • Explore counterexamples in topology that illustrate the relationship between convexity and homeomorphism.
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Mathematicians, students of topology, and anyone interested in the foundational concepts of convexity and its implications in topological spaces.

Tomer
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Hello!
It's not really a homework problem, but it should be able to help me with something.
I was just wondering: if two sets are Homeomoprh (topologically), and one of them is convex, does it mean that the other one is convex as well?

Thanks a lot!

Tomer.
 
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I just realized it probably isn't :)
 

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