  One consequence of the cosmic censorship conjecture is that any topological
structure will ultimately collapse to within the horizons of a set of black
holes, and as a result, an external classical observer will be unable to probe
it. However a single two-level quantum system (UDW detector) that remains
outside of the horizon has been shown to distinguish between a black hole and
its associated geon counterpart via its different response rates. Here we
extend this investigation of the quantum vacuum outside of an $\mathbb{RP}_2$
geon by considering the entanglement structure of the vacuum state of a quantum
scalar field in this spacetime, and how this differs from its BTZ black hole
counterpart. Employing the entanglement harvesting protocol, where field
entanglement is swapped to a pair of UDW detectors, we find that the
classically hidden topology of the geon can have an appreciable difference in
the amount of entanglement harvested in the two spacetimes for sufficiently
small mass. In this regime, we find that detectors with a small energy gap
harvest more entanglement in the BTZ spacetime; however as the energy gap
increases, the detectors harvest more entanglement in a geon spacetime. The
energy gap at the crossover is dependent on the black hole mass, occurring at a
lower values for lower masses. This also impacts the size of the entanglement
shadow, the region near the horizon where the detectors cannot harvest
entanglement. Small gap detectors experience a larger entanglement shadow in a
geon spacetime, whereas for large gap detectors the shadow is larger in a BTZ
spacetime.
