Space Suggestion: R^3

[felixpernegger]

This isnt a totally serious suggestions, but due to the obvious importance of this space we should probably have it.

However I mainly mention this, since we actually cant distinguish R^2 for R^3 currently, see the manual added traits at https://topology.pi-base.org/spaces/S000176.

I think pibase should probably be able to do this, so if someone has an idea for a suitable property distinguishing those spaces, it may be worth adding.

[GeoffreySangston]

As part of background for this MSE question, I investigated two versions of "simply connected at infinity" which appear in literature. Neither holds for $\mathbb{R}^2$ (one fails for $\mathbb{R}$ and the other doesn't, but we can still distinguish $\mathbb{R}$ and $\mathbb{R}^2$), but both hold for $\mathbb{R}^3$. Either of these properties would also allow us to distinguish the Whitehead manifold from $\mathbb{R}^3$, so we'd be able to add that famous example (though I think the Whitehead manifold would be indistinguishable from $\mathbb{R}^2$ in pi-base until additional properties were added... then again we are adding Euclidean I think).