Alexandroff compactification for closed right half plane

Question

As title says, I want to find a subspace of $\mathbb{R^2}$ or $\mathbb{R^3}$ homeomorphic to the Alexandroff compactification of $$X = \{(x,y) \in \mathbb{R^2} \colon x \geq 0\}$$ I have been trying to find a homeomorphism between $X$ and a subspace of $\mathbb{R^2}$ or $\mathbb{R^3}$ with a "clearer" Alexandroff compactification. For example, since $\mathbb{R^2}$ is homeomorphic to the unit sphere without the north pole, $S^2 \setminus N$, I deduced that $X = [0,\infty) \times \mathbb{R}$ is homeomorphic to the upper (closed) hemisphere of $S^2 \setminus N$, so the Alexandroff compactification of $X$ would be that hemisphere including the north pole. I have doubts about whether the previous homeomorphism is correct or not.

Answer 1

Maybe it's easier to identify the compactification of $X$ with the "eastern hemisphere" $E \subset S^2$, especially once you've identified the compactification of $\mathbb{R}^2$ with $S^2$, since the right half plane is taken to $E \setminus N$.