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A:
A minor but important, fix:
I’ve tried to assign my email to Gmail (only in italian) and, on every single occasion, the Gmail just showed me an italian version of the GMail help page.
I changed this to:
La mia mail è associata alla funzione di posta in Google.
(I’m not sure which the exact translation should be, but it doesn’t seem to matter what you translate to. Google is the only one capable of handling internationalized emails.)
Q:
Quotients of $\Bbb R^n$
Let $X$ be a Hausdorff topological space. Then it’s a well known fact that $X$ is homeomorphic to a quotient of $\Bbb R^n$ for some $n$. But for instance a countable power of $\Bbb R$ is not homeomorphic to any quotient of a topological vector space. Any hints for a more or less constructive proof?
A:
If $X$ is not homeomorphic to $\mathbb R^n$ then there exists $U\subseteq X$ an open subset of $X$ so that no continuous maps $\mathbb R^n\to X$ maps $U$ to $X\setminus U$.
But then the identity on $\mathbb R^n$ is a continuous map $X\to X$ so by the homotopy extension property it extends to a continuous map $X\times I\to X$. If $X\times I$ was homeomorphic to $\mathbb R^n$ then it would induce a homeomorphism $X\to \mathbb R^n$.
And as we have that $X\times I$ is compact then this map has to be surjective. This means that for every open subset $U\subseteq X$ the set $\{x\in X:x\in U\text{ or }x
otin U\}$ is a union of components of $X$ of the form $X\setminus O$ where $O$ is open in $\mathbb R^n$.
This contradicts the fact that the space $X$ is Hausdorff.
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PHP-MYSQL Connect Without using Active-Record Class
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