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Question: Construct a simply connected covering which a subspace of $\mathbb R^3$ of union of a sphere and a circle intersecting in two points.

My idea: First of all note that union of a sphere and a circle intersecting in two points is homotopy equivalent to $S^2$ wedge figure 8, then take the simply connected covering for figure 8 (which is tree) and then insert a $S^2$ everywhere the point of intersection.

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