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Let $X$ be a path connected scheme (we have an argument that this is implied by connected and Noetherian). When is the topological fundamental group of $X$ trivial? This is straightforward for certain special cases like that of an integral scheme, but we haven't been able to come up with a more general characterization.

forget this
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    Related: https://math.stackexchange.com/questions/2701914/connected-non-contractible-schemes – Eric Wofsey Oct 25 '18 at 04:14
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    Note also that if $K$ is a compact Hausdorff space, then $\operatorname{Spec} C(K)$ is homotopy equivalent to $K$. So, there's no hope of any simple answer in complete generality. – Eric Wofsey Oct 25 '18 at 04:40
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    If you work over $\mathbb{C}$ and know about the etale topology, it is a fact that the pro-finite completion of the etale fundamental group is isomorphic to the topological fundamental group (see Milne for a proof of this). So in a more general context, you could define simply connected as the vanishing of the etale fundamental group (since the etale topology makes sense much more generally). I do not know a necessary/sufficient criterion for this to happen though. – DKS Oct 25 '18 at 13:07

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