# Does there exist a local homeomorphism between these two set?

We have to decide if there is a local homeomorphism between $\mathbb{R}^{4}$ and (X$\times$Z), where X=$\mathbb{C}^{2}$ with metric topology and Z=$\mathbb{C}^{2}$ with cofinite topology. Here is my attempt at it:

If there was a local homeomorphism between g: $\mathbb{R}^{4}$$\rightarrow(X\timesZ) then for x_{0} in \mathbb{R}^{4}, there will be an open set U \subseteq$$\mathbb{R}^{4}$ such that g(U) is homeomorphic to U. However, U in metric topology is Hausdorff whereas g(U)$\subseteq$(X$\times$Z) can not be Hausdorff because (X$\times$Z) is not Hausdorff.

I also have a separate proof that since Z is not Hausdorff in cofinite topology, (X$\times$Z) is not Hausdorff. I would really appreciate any comments on this.

• correct correct correct – Forever Mozart Apr 28 '16 at 4:55
• thank you very much @foreverMozart – HumbleStudent Apr 28 '16 at 15:09