Two definitions of locally compact

[Fredrik]

I'm trying to understand the proof of (ii)$\Rightarrow$(i) of proposition A.6.2.(1) here. The theorem says that the given definition of "locally compact" is equivalent to a simpler one when the space is Hausdorff. I found the proof quite hard to follow. After a few hours of frustration I'm down to one last detail. Why is $F\subset U_1$? It seems to me that F could contain limit points of $U_1$ that aren't in $U_1$.

Edit: I figured it out. The set $V_2$ is open in the topology of $K_x$, so there's an open set $V_2'$ such that $V_2=K_x\cap V_2'$. This implies that

$$F=K_x-V_2=K_x-(K_x\cap V_2')=K_x-V_2'.$$

We also have $K_x-U_1\subset V_2\subset V_2'$, and this implies that

$$K_x-V_2'\subset K_x-(K_x-U_1)=K_x\cap U_1\subset U_1.$$

 

[fresh_42]

Thanks for posting the answer.