Which properties are preserven when taking an inverse limit?

Question

I'm studying the $p-$adic solenoid, which is defined as an inverse limit of circles with bonding maps $z^p$, and is denoted by $\mathcal{S}_p$.

I have found the following analogies:

• $S^1$ is compact, and so is $\mathcal{S}_p$
• $S^1$ is connected, and so is $\mathcal{S}_p$
• $S^1$ is an abelian topological group and so is $\mathcal{S}_p$
• $S^1$ has finite closed subgroups of every size and $\mathcal{S}_p$ has finite closed subgroups of evry size relatively prime with p.

so I wonder if there is some kind of properties that are known to be preserved when taking an inverse limit of objects who had the same property.

Answer 1

The following facts are well-known:

(1) The inverse limit of a system of compact spaces is compact. See [1] Theorem 3.2.13

(2) The inverse limit of a system of compact connected spaces is connected. See [1] Theorem 6.1.18. For non-compact spaces it is in general false.

(3) The inverse limit of a system $\mathbf{G} = (G_\alpha,p_\alpha^\beta)$ of (abelian) topological group is an (abelian) topological group. In fact, $\varprojlim \mathbf{G}$ is a subgroup of $P = \prod_\alpha G_\alpha$ which is an (abelian) topological group. It therefore inherits the structure of an (abelian) topological group from $P$.

(4) Your last point does not have a general answer. If all $G_\alpha$ have finite subgroups of every size (which are automatically closed if topological groups are required to be $T_0$), then it depends on the bondings what can be said about $\varprojlim \mathbf{G}$.

[1] Engelking, Ryszard. "General topology." (1989).