# (topological space) Manifold vs Euclidean space

#### highmath

##### Member
What is the difference between Manifold space to Euclidean space?

What properties does the Manifold space have that Euclidean space doesn't have?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
What is the difference between Manifold space to Euclidean space?

What properties does the Manifold space have that Euclidean space doesn't have?
A manifold is an abstraction of Euclidean space, so it has less properties, and allows for more.
That is, until we add extra properties to a manifold that are not possible in Euclidean space.

A manifold is only locally homeomorphic (has a bi continuous function) to Euclidean space $\mathbb R^n$.
Homeomorphic means that there is a continuous 1-1 mapping that has an inverse that is also continuous.
It means that a manifold does have angles and distances, but these can change from point to point in a continuous fashion.
As such it allows:
• Curved spaces since the global parallel postulate of Euclidean geometry is not (necessarily) true.
• Non-orientable shapes like the Klein bottle.
• Disconnected components that may even have different dimensions.