Recently, I was asked to calculate the fundamental group of the space $X= \{a,b,c,d\}$ with open sets generated by $\{ a, c, abc, acd \}$.

Turns out, $\pi_1(X)\cong \mathbb Z$ and in fact, $X$ is the quotient of $S^1$ (with the northern and southern hemispheres identified). But the result was not so easy to prove and this motivates the questions:

Is the fundamental group of every connected manifold (other restrictions / generalizations on the manifold are welcome) the fundamental group of a finite space? (Of course, it would not be Hausdorff). (I observe that there are many redundant points on a manifold where homotopy-equivalent loops need not traverse)

Is calculating $\pi_1$ of such finite spaces easier than for the given space? (If yes, this gives a method to calculate fundamental groups of many familiar spaces)

Perhaps the fact that -- $\pi_1$ of any CW complex just depends on its 1-skeleton [edit:2 skeleton]-- may be helpful.

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