A topological space $Y$ has the universal extension property if for every normal space $X$, every closed subset $A$ of $X$, and every continuous function $f:A\rightarrow Y$, we can extend $f$ to a continuous function $g:X\rightarrow Y$. Now for any $J$, the product space $\mathbb{R}^J$ has the universal extension property, and so does every retract of $\mathbb{R}^J$.
But my question is, what is an example of a topological space which has the universal extension property but is not homeomorphic to $\mathbb{R}^J$ or any of its retracts? Or does no such example exist?
Best Answer
Your question is whether any $Y$ which has the universal extension property admits a closed embedding into some $\mathbb{R}^J$.
The answer is "no". Let $Y$ have more than one point and the trivial topology. Then it has the universal extension property, but cannot embedded into any $\mathbb{R}^J$.
To obtain a closed embedding, you therefore need additional assumptions on $Y$. For example, if $Y$ is compact, then it embeds into some Tychonoff cube $[0,1]^J$ which is contained in $\mathbb{R}^J$.