Suppose that a topological space $X$ is the union of an increasing sequence of open subsets each of which is homeomorphic to the Euclidean space $R^n$. Then $X$ itself is homeomorphic to $R^n$. This is an old theorem of Morton Brown. We observe that this theorem is an immediate consequence of other two theorems of Morton Brown concerning near homeomorphisms and cellular sets.
When is $X\times Y$ homeomorphic to $X\times_l Y$?
Short proof of a theorem of Juhasz
On metrizable $X$ with $C_p(X)$ not homeomorphic to $C_p(X)\times C_p(X)$
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