What kind of "direct product" is the $p$-adic solenoid? I'm confused about this paragraph in https://en.wikipedia.org/wiki/P-adic_number:

$p$-adic integers can be extended to $p$-adic solenoids $\mathbb{T}_{p}$ in the same way that integers can be extended to the real numbers, as the direct product of the circle ring $\mathbb{T}$ and the $p$-adic integers $\mathbb{Z}_{p}$.

I don't think the real numbers are the kind of topological product (or the kind of group product) that I'm used to of the integers and the circle, and neither is the $p$-adic solenoid the usual kind of product of the $p$-adic integers and the circle. Is there a more specific name for this kind of product, and what is it?
Only looking at the topological structure, the only relationship I can see is this one:

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*There's a map $\pi_{\mathbb R} : \mathbb R \to \mathbb T$ such that for each point $x \in \mathbb T$, $\pi_{\mathbb R}^{-1}(x)$ is homeomorphic to $\mathbb Z$ and furthermore there is some open neighborhood $O$ of $x$ such that $\pi_{\mathbb R}^{-1}(O)$ is homeomorphic to $O \times \mathbb Z$.

*There's a map $\pi_{\mathbb T_p} : \mathbb T_p \to \mathbb T$ such that for each point $x \in \mathbb T$, $\pi_{\mathbb T_p}^{-1}(x)$ is homeomorphic to $\mathbb Z_p$ and furthermore there is some open neighborhood $O$ of $x$ such that $\pi_{\mathbb T_p}^{-1}(O)$ is homeomorphic to $O \times \mathbb Z_p$.

But I don't know anything about what this kind of topological product is called, nor how to define this kind of product for topological groups or rings.

This construction is the inverse limit of $\mathbf R/p^k\mathbf Z$ over all nonnegative integers $k$.  Since the appendix to chapter 1 of A. Robert's "A Course in p-adic Analysis" for details.
