Hausdorff condition in differential manifold definition

Thread starter Goldbeetle
Start date Jul 2, 2010
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Condition Definition Differential Manifold

In summary, the Hausdorff condition is a topological property that is important in defining a differential manifold. It ensures that the manifold is well-behaved and locally Euclidean, making it useful for calculations and applications in physics and engineering. While it is one of several conditions that define a differential manifold, it is typically included due to its significance. However, there are alternative definitions that do not require the Hausdorff condition, but they are less commonly used.

Jul 2, 2010

Goldbeetle

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

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Jul 2, 2010

zhentil

The line with two origins is locally Euclidean, but not Hausdorff.

Jul 2, 2010

Goldbeetle

OK, true.

1. What is the Hausdorff condition in the definition of a differential manifold?

The Hausdorff condition is a topological property that states that any two distinct points in a space must have disjoint neighborhoods. In the context of a differential manifold, this means that for any two points on the manifold, there exist open sets around each point that do not overlap.

2. Why is the Hausdorff condition important in the definition of a differential manifold?

The Hausdorff condition ensures that a differential manifold is a well-behaved space, allowing for smooth and consistent calculations and measurements. It also guarantees that the manifold is locally Euclidean, which is a key property for many applications in physics and engineering.

3. How does the Hausdorff condition differ from the other conditions in the definition of a differential manifold?

The Hausdorff condition is one of several conditions that, together, define a differential manifold. The other conditions include being a topological space, being locally Euclidean, and having a maximal atlas. The Hausdorff condition specifically ensures that the manifold is a well-behaved topological space.

4. Can a differential manifold fail to satisfy the Hausdorff condition?

Yes, it is possible for a space to fail the Hausdorff condition and still satisfy the other conditions of a differential manifold. However, this would make the space more difficult to work with and would limit its applications. Therefore, the Hausdorff condition is typically included in the definition of a differential manifold.

5. Is the Hausdorff condition always necessary for a space to be considered a differential manifold?

No, there are some alternative definitions of a differential manifold that do not require the Hausdorff condition. However, these alternative definitions are less commonly used and may have different properties and applications. In most cases, the Hausdorff condition is considered an essential aspect of a differential manifold.