# Can you modify solenoids to be locally connected?

Solenoids are not locally connected. Intuitively, this is because if you look at a neighborhood around a point, the other "strands" will be in the neighborhood, since there are infinitely many strands arbitrarily close to every point. You could say that regular solenoids are "wrapped" too tightly.

My question is can you "unwrap" the solenoid so as to make it locally connected. The relationship between the "unwrapped" solenoid and the regular solenoid would be similar to that of the relationship between the rose with infinite petals and the Hawaiian earring.

In particular, the "unwrapped" solenoid would be strictly finer than the regular solenoid, just as the infinite rose is strictly finer than the Hawiian earring.

Sure. If you want to "disentangle" a small piece of arc (a "strand") from all the other strands around it, just declare it to be open. In other words, you can refine your solenoid by declaring every homeomorphic copy of $(0,1)$ to be an open set.
The resulting topological space is just a $\mathfrak c$-sized disjoint union of copies of $\mathbb R$. Each copy corresponds to a composant of the original solenoid.