# Topology of complex C^n space

1. Jun 28, 2010

### facenian

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

2. Jun 28, 2010

### Hurkyl

Staff Emeritus
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)

3. Jun 28, 2010

Thank you!