Are All Manifolds Defined to Be Hausdorff?

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SUMMARY

All manifolds are defined as Hausdorff spaces, which is an integral part of the definition of a topological manifold. This requirement is not merely an assertion but is established within the framework of topology. While there are non-Hausdorff spaces that can be locally homeomorphic to Euclidean spaces, they do not qualify as manifolds under the standard definition. The Hausdorff property is essential for the structure and properties of manifolds.

PREREQUISITES
  • Understanding of topological spaces
  • Familiarity with the concept of Hausdorff spaces
  • Knowledge of manifolds and their definitions
  • Basic grasp of local homeomorphism in topology
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  • Study the definition and properties of topological manifolds
  • Explore the implications of the Hausdorff condition in topology
  • Investigate examples of non-Hausdorff spaces and their characteristics
  • Learn about local homeomorphisms and their role in manifold theory
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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 properties.

hideelo
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Is the fact that all manifolds are hausdorff spaces a part of the definition, or can this be proven from the fact that it is a set which is locally isomorphic to open subsets of a hausdorff space?

P.S. if it can be proven I don't want to know the proof, I want to keep working on it, I just want to know that hausdorff property is not some assertion put in from the outset.
 
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The Hausdorff requirement is part of the definition of a topological manifold. There exist non-Hausdorff spaces which are locally homeomorphic to Euclidean spaces.
 
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