Example of a non locally-compact space with a dense locally compact subspace. Let $X$ be a topological space such that $A \subset X$ is a dense subspace which is locally compact and $B \subset X$ is a dense subspace which is not locally compact (at all of its points).
Is it possible to find such $X$? if it is, an example?

1)If $X$ must be Hausdorff, take $X=\mathbb{R}$, $A=X-\{0\}$, $B=\mathbb{Q}$.
2)If $X$ must be a non-locally compact Hausdorff space, let $Y=\{1/n|n\in \mathbb{N}^*\}$ and $X=(\{0\}\times \mathbb{Q}) \cup (Y\times \mathbb{R})$, $X$ is not locally compact at the point $(0,0)$.
Let $A=Y\times \mathbb{R}$, $A$ is dense in $X$ and is locally compact.
Let $B=Y \times \mathbb{Q}$, $B$ is dense in $X$ and $B$ is not locally compact at all of its points.
