The Moore Plane's topology In the definition of the Moore plane $X=L{_1}\cup L{_2}$, where $L{_1}$ is the line $y=0$ and
$L{_2}=X\setminus L{_1}$ , I have a problem. In the Engelsking's book, for each $x\in L{_1}$
neghbourhood of $x$, is the form $U(x,1/i)\cup \{ x \}$ where $U(x,1/i)$ be the set of $X$ inside the circle centered $x$ and radius $1/i$ for i=1,2,..... So, I wonder that whether or not radius is greater than $1$?If it is, how can cover all $X$ with small radius? thanks

You don’t need to cover $X$ with basic open nbhds of points of $L_1$: you also have the basic open nbhds of points of $L_2$, which are ordinary Euclidean balls small enough to stay within $L_2$. Specifically, the following collection is a base for $X$:
$$\left\{\{\langle x,0\rangle\}\cup B\left(\left\langle x,\frac1k\right\rangle,\frac1k\right):k\in\Bbb Z^+\right\}\cup\left\{B(\langle x,y\rangle,\epsilon):y>0\text{ and }0<\epsilon\le y\right\}\;,$$
where for $B(p,r)$ is the usual Euclidean open ball of radius $r$ centred at $p$.
Note that your description of basic open nbhds at points of $L_1$ isn’t actually correct: the ball $U$ is tangent to $L_1$ at $x$ and therefore does not have its centre at $x$.
