Pauli's exclusion principle forces fermions to occupy distinct quantum states, creating a filled region of momentum space at low temperature, the Fermi sea, whose topology governs the system's response to perturbations and the nature of its correlation functions. Recent theory predicts that for non-interacting fermions, the Euler characteristic of a
D-dimensional Fermi sea -- the topological invariant that describes its shape -- is encoded in its (
D+1)-point density correlations. Here we experimentally demonstrate this connection in a two-dimensional degenerate gas of neutral
6Li atoms using single-atom-resolved imaging. By measuring three- and four-point connected density correlations in real space, we directly extract topological invariants of the underlying Fermi sea, including the Euler characteristic. Our results are in remarkable agreement with ideal-gas predictions, despite the presence of sizeable interactions, and establish a new pathway for probing many-body topology through correlation measurements.