Separating projective three-space in a positive connected sum

Tube the projective lines in the two summands of ${\mathbb{CP}}^2\#{\mathbb{CP}}^2$ through the connecting neck. Their connected sum is an embedded two-sphere in the class $h_1+h_2$, with self-intersection two. Its oriented normal disk bundle has Euler number two, so its boundary is the unit tangent bundle of $S^2$, namely $SO(3)\cong {\mathbb{RP}}^3$. The boundary separates the interior of the tubular neighborhood from its nonempty exterior.

// Destination: geometry-and-topology.bigb