In their topology book, Fuchs and Fomenko define the Tychonoff space $T$ to be the set of all real sequences $(x_1, x_2, x_3, ...)$ with the base of topology formed by the sets $\{(x_1, x_2, x_3, ...) \in T \ | \ (x_1, ..., x_n) \in U\}$ for all $n$ and all open $U \subset \mathbb{R}^n$. I'm not quite sure what the basis elements are here. So a generic element of the base is a real sequence $(x_1, x_2, x_3, ...)$ such that it's first $n$ terms $(x_1, ..., x_n)$ are in $U$ for an open $U$. But it says that for all $U$ open in $\mathbb{R}^n$. So $(x_1, ..., x_n) \in U$ for every open $U \subset \mathbb{R}^n?$ But that's nonsense obviously. So do we fix an open set $U \subset \mathbb{R}^n$ for every $n$, first? If we do, then what happens to the claim "for all open subsets $U \subset \mathbb{R}^n$?
I would truly appreciate some help here to clarify what is going on. Thanks!