Find compactification of $\mathbb{R}$ which has a subset homeomorphic to $\mathbb{R}^2$

Consider $X=\mathbb{R}$ with the standard topology.How can I find a compactification $Y$, such that $Y$ is (of course) compact, hausdorff and has a subset which is homeomorphic to $\mathbb{R}^2$?

This question is a part of a bigger question, where in the first bullets I proved that $$ \left\{ \left(t,\sin\left(\frac{1}{1-t^{2}}\right)\right)\in\mathbb{R}^{2}\thinspace:\thinspace t\in\left(-1,1\right)\right\} \cup\left(\left\{ -1,1\right\} \times\left[-1,1\right]\right) $$

Is a compactification of $\mathbb{R}$. Can it be utilized in order to deduce a compactification as I stated above?

Thanks in advance.