= Fundamental group of a sphere with an attached chord
{title2=$\pi_1(S^2\cup J)\cong\mathbb Z$}
Attach a <closed interval> to a <sphere> $S^2$ at two distinct endpoints, with its interior disjoint from the <sphere>. The <fundamental group> is <infinite cyclic group>. Remove a point of the <sphere> away from the endpoints: the remaining plane has a <deformation retraction> onto an <arc> between the endpoints, keeping the added <closed interval> fixed. This gives a <circle>. An open cap about the removed point is <simply connected>, and its punctured overlap has trivial map into the first <fundamental group> because it lies in the plane. The <Seifert-van Kampen theorem> therefore adds no further <relator>.
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