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

Topology of complex C^n space

  1. Jun 28, 2010 #1
    How is the topology in [itex]C^n[/itex] defined? is it correct to think of it like this:
    suppose the biyective map [itex]h:C^n\rightarrow R^{2n}[/itex] given by [itex]h[(z_1,\ldots,z_n)]=(x_{11},x_{12},\ldots,x_{n1},x_{n2})[/itex] where [itex]z_i=(x_{i1},x_{i2})[/itex] then the topology of C^n is defined by declaring h to be an isometry.
  2. jcsd
  3. Jun 28, 2010 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    It's the product topology on the Cartesian product of n copies of C.

    Of course, the function you wrote does induce a homeomorphism between the standard topologies on Cn with R2n.

    (and this does, in fact, turn out to be an isometry of the standard metric space structures on these two sets as well)
  4. Jun 28, 2010 #3
    Thank you!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook