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Suppose that a topological space $X$ is the union of an increasing sequence of open subsets $U_i$c each of which is homeomorphic to the Euclidean space $\mathbb{R}^n$. How does one show that $X$ is homeomorphic to $\mathbb{R}^n$? I am unable to prove it in the case $n=2$. This is Dieudonne's problem.

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This should answer your question:

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