{title2=$H^1(\mathbb R^2\setminus\{(n,0):n\geq1\};\mathbb Z)\cong\prod_{n\geq1}\mathbb Z$}

The connected open set $X=\mathbb R^2\setminus\{(n,0):n\geq1\}$ retracts onto a locally finite graph having one independent loop around each puncture. Hence $H_1(X;\mathbb Z)\cong\bigoplus_{n\geq1}\mathbb Z$, while the <universal coefficient theorem for cohomology> gives
$$
H^1(X;\mathbb Z)\cong\operatorname{Hom}\left(\bigoplus_{n\geq1}\mathbb Z,\mathbb Z\right)\cong\prod_{n\geq1}\mathbb Z,
$$
an uncountable group.


 First cohomology of the countably punctured plane

ID: first-cohomology-of-the-countably-punctured-plane