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

[sadegh4137]

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.

 

[micromass]

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 \mathbb{R}^n.

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

2) M is Hausdorff

3) M is second countable

 

[morphism]

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.

 

[sadegh4137]

what s the meaning of second countable?

 

[Matterwave]

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.