Countable Connected Hausdorff Space What is an example of a connected Hausdorff space with a countable number of elements? Any finite Hausdorff space is discrete (therefore not connected) and there are a myriad of examples uncountable Hausdorff spaces. But what about countable?
 A: Quite a few examples have been constructed since Uryson produced the first one in $1925$ (see this MO post for a reference). One of the simplest, sometimes called Bing’s sticky foot space, was published by R.H. Bing in $1953$.
The points of the space $X$ are the points $\langle p,q\rangle\in\Bbb Q^2$ such that $q\ge 0$. Let $Y=\Bbb Q\times\{0\}$. For each $\epsilon>0$ and $x=\langle q,0\rangle\in Y$ let $B(x,\epsilon)=\{\langle p,0\rangle\in Y:|p-q|<\epsilon\}$. For each $x=\langle p,q\rangle\in X\setminus Y$ let $T_x$ be the equilateral triangle in $\Bbb R^2$ with one vertex at $x$ and the other two on the $x$-axis; the two base vertices have the irrational $x$-coordinates $\ell(x)=p-\frac{q}{\sqrt3}$ and $r(x)=p+\frac{q}{\sqrt3}$, so they don’t actually lie in $X$. For $\epsilon>0$ let 
$$B(x,\epsilon)=\{x\}\cup\{\langle s,0\rangle\in Y:|s-\ell(x)|<\epsilon\text{ or }|s-r(x)|<\epsilon\}\;.$$
For both kinds of point $\{B(x,\epsilon):\epsilon>0\}$ is a local base at $x$.
If you think of $\ell(x)$ and $r(x)$ as the left and right ‘feet’ of the point $x$, you can see where the name sticky foot comes from: open nbhds of $x$ are little clots of points on $Y$ ‘stuck to’ the feet of $x$.
Clearly $X$ is countable, and it’s easy to check that $X$ is Hausdorff. To see that $X$ is connected, observe that for any $x,y\in X$ and $\epsilon,\delta>0$ we have $\operatorname{cl}B(x,\epsilon)\cap\operatorname{cl}B(y,\delta)\ne\varnothing$. (A picture helps here: once you see what the closures of these open sets are, the result is very clear.) Thus, no two non-empty open sets have disjoint closures, and it follows easily that $X$ is connected.
