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Does there exist a finite topological space with fundamental group isomorphic to $\mathbb{Z_2}$?

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Yes. In the paper "Singular homology groups and homotopy groups of finite topological spaces" by Michael McCord [Duke Math. J. 33 (1966) pp. 465-474], it is proved (among other things) that every finite simplicial complex $K$ has its geometric realization weakly homotopy equivalent to a finite topological space. Apply that with $K$ being a triangulation of the projective plane.

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