compactification of non-compact metrizable space

Question

I want to try to represent the uncountable product of unit interval $[0,1]$ as a Hausdorff compactification of non compact metrizable space $X$. I need a homeomorphism from $X$ to uncountable product. Could you give me any hint?

thanks,

Answer 1

For each $\alpha\in\Bbb R$ let $I_\alpha$ be a copy of $[0,1]$, and let $X=\prod_{\alpha\in\Bbb R}I_\alpha$. Let $Q=\Bbb Q\cap[0,1]$. For each finite $F\subseteq\Bbb Q$ and function $\varphi:F\to Q$ define $x(F,\varphi)\in X$ as follows. Let $F=\{q_1,\dots,q_n\}$, where $q_1<\ldots<q_n$; then for each $\alpha\in\Bbb R$

$$x(F,\varphi)_\alpha=\begin{cases} 0,&\text{if }\alpha<q_1\\ \varphi(q_k),&\text{if }q_k\le\alpha<q_{k+1}\text{ for some }k\in\{1,\dots,n-1\}\\ \varphi(q_k),&\text{if }\alpha\ge q_n\;. \end{cases}$$

Let $Y=\{x(F,\varphi):F\subseteq\Bbb Q\text{ is finite and }\varphi:F\to Q\}$; clearly $Y$ is countable.

• Show that $Y$ is dense in $X$.

Now let $Z=\prod_{q\in\Bbb Q}I_q$; clearly $Z$ is metrizable. Let $\pi:X\to Z$ be the obvious projection map.

• Show that $\pi[Y]$ is homeomorphic to $Y$ and hence that $Y$ is metrizable.