Is there a countable connected space $(X,\tau)$ such that for all $x\in X$ the space $X\setminus\{x\}$ is not connected any more with the induced subspace topology?



Countable connected space where removing $1$ point destroys connectedness
Let $\mathbb{R}$ be with its usual topology, and let $f:\mathbb{R}\to \mathbb{Z}$ defined by:
$$
f(x)=\left\{\begin{matrix}
2k & x=2k,\text{ where } k\in\mathbb{Z}\\ 
2k+1 & 2k<x<2k+2,\text{ where } k\in\mathbb{Z}
\end{matrix}\right.
$$
Let $X$ be $\mathbb{Z}$ with the quotient topology induced by $f$. 


