In Topology, the second edition by Munkres, in section 19, on page 113 he says the following:
"So let us consider the cartesian product
\begin{align*}
X_1\times …\times X_n \quad and \quad X_1\times X_2 \times …,
\end{align*}
where each $X_i$ is a topology space. There are two possible ways to proceed. One way is to take as basis all sets of the form $U_1\times …\times U_n$ in the first case, and of the form $U_1\times U_2 …$ in the second case, where $U_i$ is an open set of $X_i$ for each $i$. This procedure does indeed define a topology on the cartesian product; we shall call it the box topology.
Another way to proceed is to generalize the subbasis formulation of the definition, given in §15. In this case, we take as a subbasis all sets of the form $\pi_i^{-1}(U_i)$, where $i$ is any index and $U_i$, is an open set of $X_i$,. We shall call this topology the product topology.
How do these topologies differ? Consider the typical basis element $B$ for the second topology. It is a finite intersection of subbasis elements say for $i = i_1,…, i_k$. Then a point $\mathbf{x}$ belongs to $B$ if and only if $\pi_i(\mathbf{x})$ belongs to $U_i$, for $i = i_1,…,i_k$; there is no restriction on $\pi_i(\mathbf{x})$ for other values of $i$."
My question is:
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Why $i = i_1,…, i_k$? I think it should $i=1,…,k$.
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I don't understand "there is no restriction on $\pi_i(\mathbf{x})$ for other values of $i$".
Can someone help me? Thanks.
Best Answer
The subbase is all sets of the form $\pi_i^{-1}[U_i]$ where $i \in I$ (this can be any index set not just $\Bbb N$ BTW) and $U_i \subseteq X_i$ is open.
The generated subbase thus takes finitely many indices and open sets and not necessarily the first $k$ (if there even are “first indices”, $I$ could be $\Bbb Z$ or $\Bbb R$ or even larger sets) or “consecutive” ones (same remarks on arbitrariness of index sets apply). Hence the notation $i_1,\ldots, i_k$, it’s just a way of showing that the finite set is from the index set $I$ (hence small $i$) with subscripts to distinguish and count them.
And if $x$ in the product $\prod_{i \in I} X_i$ (a more neutral way of denoting it then the only “$\Bbb N$-suggesting” $X_1 \times X_2 \times \ldots X_n \times \ldots$) is in the set $$B=\pi_{i_1}^{-1}[U_{i_1}] \cap \ldots \pi_{i_k}^{-1}[U_{i_k}]$$ iff we have $x_{i_j} = \pi_{i_j}(x) \in U_{i_j}$ for all $j=1,\ldots k$ so we only have a condition on the finitely many coordinates $\{i_1,\ldots,i_k\}$ and no idea what $x_i$ could be for all other (infinitely many, usually) $i \in I$, only that it’s in $X_i$ by virtue of being in the product $\prod_{i \in I} X_i$.
Hope that clarifies and elucidates Munkres’ discussion. In the beginning Munkres only treats finite products and countable ones indexed by $\Bbb N$ but later on he will discuss the most general case, as I just did.