Is there an infinite, countable connected $T\_2$-space $(X,\tau)$ such that $(X,\tau)$ has the fixed point property? (This means that for every continuous map $f:X\to X$ there is $x\in X$ such that $f(x) = x$.)



Countably infinite connected Hausdorff space with the fixed point property
Yes, there is such an example.


This is nearly Problem 10705 in **The American Mathematical Monthly**, proposed by D. W. Brown in [**106** #1 (January 1999), p. 67](http://www.jstor.org/stable/2589591), where the problem asks for a countably infinite $T\_{2}$ example. In an editorial comment following John Cobb's solution in [**107** #4 (April 2000), pp. 375-376](http://www.jstor.org/stable/2589200), it is stated that Prabir Roy's lattice space --- [*A countable connected Urysohn space with a dispersion point*](https://projecteuclid.org/euclid.dmj/1077376386), **Duke Mathematical Journal** 33 #2 (1966), pp. 331-333 --- is an example that is a countable *connected* Urysohn space. For what it's worth, my notes on this problem say that this can be verified by making use of the analog in Roy's space of observations (a) and (b) at the beginning of the proof of the theorem at the bottom of p. 375.


