generating topology Let $X$ be the space in Engelking's book, example 5.1.23. (page 307) and $M$ closed discrete subspace of $X$( this is  also called the Bing's example)By using this, consider the 
set $Z=(M\cup \{0\})\cup \bigcup_{i=k} (X\times \{1/i\})$, and generate a topology on $Z$ taking as a base at a point $(x,0)$ the sets $\{(x,0)\}\cup  \bigcup_{i=k} (U\times \{1/i\})$, where $U$ is a neighbourhood of the point $x$ in the space $X$ and k=1,2....., and letting all the remaining points be isolated. ( see Engelking's book, page 338, exercise 5.5.3).
The topology on $Z$ is not clear for me. Could you  help me how I describe the topology?
thanks in advance,
 A: The main problem is that there’s a typo in Engelking: it should be
$$Z=\Big(M\times\{0\}\Big)\cup\bigcup_{i=0}^\infty\left(X\times\left\{\frac1i\right\}\right)\;.$$
Let $D=\left\{\frac1n:n\in\Bbb Z^+\right\}$, and let $K=D\cup\{0\}$; then $Z$ is a subset and almost a subspace of $X\times K$.
Start with $X\times K$. Now throw out all of $X\times\{0\}$ except the points in $M\times\{0\}$. Make every point of $X\times D$ isolated; the only non-isolated points of $Z$ are the points $\langle x,0\rangle$ for $x\in M$. To get a basic open nbhd of $\langle x,0\rangle$ in $Z$, start with a basic open nbhd $B$ of $\langle x,0\rangle$ in $X\times K$: it will have the form $U\times V_n$, where $U$ is an open nbhd of $x$ in $X$, and $V_n=\{0\}\cup\left\{\frac1k:k\ge n\right\}$. Now throw away all of $B\cap(X\times\{0\})$ except the point $\langle x,0\rangle$ itself; what’s left is a basic open nbhd of $\langle x,0\rangle$ in $Z$. This makes $M\times\{0\}$ a closed, discrete subset of $Z$.

It might be helpful to see a similar construction in a more familiar setting. Let
$$X=\Big(\Bbb Q\times\{0\}\Big)\cup\left\{\langle x,y\rangle\in\Bbb R^2:y>0\right\}\;.$$
Make $\langle x,y\rangle\in X$ isolated if $y>0$. For $\epsilon>0$ and $x\in\Bbb Q$ let
$$B(x,\epsilon)=\{\langle x,0\rangle\}\cup\Big((x-\epsilon,x+\epsilon)\times(0,\epsilon)\Big)\;;$$
the sets $B(x,\epsilon)$ with $\epsilon>0$ are a local base at $\langle x,0\rangle$. The construction of $Z$ from $X$ is similar, with $M$ playing the part of $\Bbb Q$ here, and $X\times D$ playing the part of the open upper half-plane here.
