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 $f$.

The other answer describes the "Khalimsky line". It is not $T_1$, but it is possible to obtain Hausdorff examples by starting with a countable connected Hausdorff space $X$, blowing up its points into more copies of $X$, and continuing this process infinitely many times. This ever-branching countable "tree" of $X$'s can be topologized so that it is connected, Hausdorff, and removing any point disconnects the space.