= 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
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