The leaves of the Kronecker foliation of the torus are diffeomorphic to the real line $\mathbb{R}$ and are dense on $T^2$. I cannot, however, describe what is the leaf space $T^2/\mathcal{F}$.
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If every leaf is dense, every point in the leaf space lies in the closure of any other point. This characterises the co-discrete space.
Alexander Golys
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In a discrete space, the exact opposite of what you said holds: every point subset is open, and therefore no point lies in the closure of any other point. – Lee Mosher Jul 11 '23 at 23:14
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Obviously, I meant co-discrete – Alexander Golys Jul 11 '23 at 23:15