What is the Definition of a Manifold and How Does it Relate to Topology?

In summary: This is a useful property because it allows us to work with countable collections rather than uncountable ones.
  • #1
sadegh4137
72
0
i see the definition of differential manifolds in some book for example, NAKAHARA

but what is the definition of manifold in general!
and what the definition of topological manifold.
 
Physics news on Phys.org
  • #2
The definition of a topological manifold depends. But you always have the following:

M is a topological manifold if it is a topological space satisfying:

1) M is locally Euclidean: For every point p in M, there is an integer n>0 and an open set U of p such that U is homeomorphic to an open subset of [itex]\mathbb{R}^n[/itex].

Sometimes, we demand (some of) the following extra axioms:

2) M is Hausdorff

3) M is second countable
 
  • #3
Different people mean different things when they say "manifold". For example, a differential geometer will likely mean differential manifold (or maybe ##C^r## manifold or ...) whereas a topologist might mean topological manifold. So it's always a good idea to be aware of what type of manifold is under consideration.

As to what is a topological manifold - have you tried doing a google search? It's really easy to find a definition online, e.g. http://en.wikipedia.org/wiki/Topological_manifold

Note that a differential manifold is in particular a topological manifold.
 
  • #4
what s the meaning of second countable?
 
  • #5
A second countable topological space is one which has a countable basis. I.e. it has a countable collection of open sets such that every open set can be expressed as a union of sets in this collection.
 

What is a manifold?

A manifold is a mathematical object that is defined as a topological space that locally resembles Euclidean space. In simpler terms, it is a space that looks flat when zoomed in, but may have a more complex overall shape.

What are the different types of manifolds?

There are several types of manifolds, including smooth manifolds, topological manifolds, and differentiable manifolds. These types differ in the level of smoothness and continuity of the space.

Why are manifolds important in mathematics?

Manifolds are important in various areas of mathematics, including geometry, topology, and physics. They provide a way to study complex spaces by breaking them down into simpler, locally flat pieces.

How are manifolds used in real-world applications?

Manifolds have many practical applications, such as in computer graphics, robotics, and machine learning. They can also be used to model physical phenomena, such as the shape of the universe or the behavior of fluids.

What is the difference between a manifold and a manifold bundle?

A manifold bundle is a more complex mathematical object that consists of multiple manifolds that are connected in a specific way. In contrast, a manifold is a single, self-contained space.

Similar threads

  • Differential Geometry
Replies
11
Views
698
  • Differential Geometry
Replies
10
Views
661
  • Differential Geometry
Replies
20
Views
3K
  • Differential Geometry
Replies
1
Views
1K
  • Differential Geometry
Replies
21
Views
588
  • Differential Geometry
Replies
4
Views
2K
Replies
4
Views
2K
  • Differential Geometry
Replies
14
Views
776
  • Differential Geometry
Replies
4
Views
2K
  • Differential Geometry
Replies
2
Views
2K
Back
Top