First cohomology of the countably punctured plane ($H^1({\mathbb{R}}^2\setminus \{(n,0):n\ge 1\};\mathbb{Z})\cong {\prod }_{n\ge 1}\mathbb{Z}$)

The connected open set $X={\mathbb{R}}^2\setminus \{(n,0):n\ge 1\}$ retracts onto a locally finite graph having one independent loop around each puncture. Hence $H_1(X;\mathbb{Z})\cong {⨁}_{n\ge 1}\mathbb{Z}$, while the universal coefficient theorem for cohomology givesan uncountable group.