Consider \begin{align} X_1 &= \{(x,y,z)\in\mathbb{R}^3:(z=0) \text{ or } (x=y=0,z\geq 0)\}, \\ X_2 &= \{(x,y,z)\in\mathbb{R}^3:(z=0) \text{ or } (x=0, y^2+z^2=1, z\geq 0)\}, \\ X_3 &= \{(x,y,z)\in\mathbb{R}^3:(x^2+y^2+z^2=1) \text{ or } (y=0,z= 0, \frac{1}{2}<|x|<1)\}. \end{align}
Show that $X_1,X_2,X_3$ are locally compact, with using the basic properties of locally compact spaces. Then show that the one-point compactifications of $X_1,X_2,X_3$ are homeomorphic to each other.
I drew all three spaces and found out $X_1$ is the $xy$-plane or the positive $z$-axis. $X_2$ is also the $xy$-plane or a half circle in the yz-plane, when $z$ is positive. $X_3$ is the $S^2$-sphere or the two open intervals $(-1,-\frac{1}{2})$ and $(\frac{1}{2},1)$ on the $x$-axis.
Now I don't know how to show these spaces are locally compact. I know some space is locally compact if $\forall x\in X_i, i\in\{1,2,3\}$ there exists a compact neighborhood, but what are other basic properties and how can I use these?
Also the one-point compactification, I understand that the one-point compactification of the $xy$-plane is homeomorphic to $S^2$. I also can show that the one-point compactification of the positive $z$-axis is homeomorphic to the one-point compactification of the half circle in $X_2$. But what about the two open intervals on the $x$-axis in $X_3$?
Hope someone can help me! :)
Some ideas:
If $X$ is locally compact Hausdorff (like $\mathbb{R}^n$) then every closed subset $C$ of $X$ is also locally compact, which will take care of $X_1$ and $X_2$. Also any intersection $O \cap C$ is locally compact, if $O \subseteq X$ is open and $C \subseteq X$ is closed, which I think will cover the $X_3$ case.
Also, $X$ and $Y$ have homeomorphic one-point compactifications if we can find a compact space $Z$ such that, for some points $p,q \in Z$, $Z \setminus \{p\}$ is homeomorphic to $X$ and $Z \setminus \{q\}$ is homeomorphic to $Y$. This follows from the essential uniqueness of the one-point compactification.