Defining the Topology of C^n: Isometry and Bijective Maps

[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.

 

[Hurkyl]

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)

 

[facenian]

Thank you!