Morton Brown's theorem

Question

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.

Answer 1

This should answer your question: http://arxiv.org/pdf/math/0404372v1.pdf.