Hausdorff condition in differential manifold definition

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SUMMARY

The discussion centers on the necessity of the Hausdorff condition in the definition of differential manifolds. It is established that while a manifold's base set M is locally homeomorphic to Rn, which is Hausdorff, this does not guarantee that M itself is Hausdorff. The example of the "line with two origins" illustrates that a space can be locally Euclidean yet fail to meet the Hausdorff condition, underscoring the importance of explicitly requiring this property in manifold definitions.

PREREQUISITES
  • Understanding of differential manifolds
  • Familiarity with topological spaces
  • Knowledge of Hausdorff spaces
  • Basic concepts of local homeomorphism
NEXT STEPS
  • Research the properties of Hausdorff spaces in topology
  • Study the implications of local homeomorphism in manifold theory
  • Explore examples of non-Hausdorff spaces and their characteristics
  • Learn about the significance of the Hausdorff condition in various mathematical contexts
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Mathematicians, particularly those specializing in topology and differential geometry, as well as students seeking to deepen their understanding of manifold theory and its foundational requirements.

Goldbeetle
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Dear all,
why is it needed in the diff manifold definition that the base set M is topologically Hausdorf ?
Since M is locally homeomorphic with Rn as metric space is Hausdorf, shouldn't this condition be automatically satisfied?
Thanks.
Goldbeetle
 
Physics news on Phys.org
The line with two origins is locally Euclidean, but not Hausdorff.
 
OK, true.
 

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