Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Definition of manifold

  1. Apr 4, 2012 #1
    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.
  2. jcsd
  3. Apr 4, 2012 #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
  4. Apr 4, 2012 #3


    User Avatar
    Science Advisor
    Homework Helper

    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.
  5. Apr 6, 2012 #4
    what s the meaning of second countable?
  6. Apr 6, 2012 #5


    User Avatar
    Science Advisor
    Gold Member

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook