Is there any simple approach to prove or disprove that $C^3$ is homeomorphic to $R^6$ in Euclidean Topology?
Can anyone give me a hint? I know Topology till compactness.
2026-03-31 20:05:49.1774987549
Is $C^3$ homemorphic to $R^6$?
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Define$$\begin{array}{ccc}\mathbb{C}^3&\longrightarrow&\mathbb{R}^6\\(z_1,z_2,z_3)&\mapsto&(\operatorname{Re}z_1,\operatorname{Im}z_1,\operatorname{Re}z_2,\operatorname{Im}z_2,\operatorname{Re}z_3,\operatorname{Im}z_3).\end{array}$$It is a continuous bijection, right?! Can you prove that the inverse is continuous too?