Confusion on the definitions of product topology

general-topologyproducts

All the books I have read so far defined the product topology $\tau$ on the finite Cartesian product of spaces $(X_i, \tau_i)_{i \le n}$ for some $n \in \mathbb{N}$ by the topology generated by the basis
$$
B = \{\prod_{i \le n} U_i ~|~ \forall i \le n: U_i \in \tau_i\}
$$
(Is the generated topology called the box topology?)

But I recently encountered a weird definition on ProofWiki, where the topology is defined by using the canonical projection maps:
$$
\tau
=
\{ \pi_i^{-1}[U] ~|~ i \le n,~ U \in \tau_i\}
$$
to make the preimages of open subsets of each space also open in the product space.

But in my knowledge, the preimage of $U$ under $\pi_i : \prod_k X_k \to X_i$ is
$$
X_1 \times \cdots \times X_{i-1} \times U \times X_{i+1} \times \cdots \times X_n
$$
that looks like a cylinder.
I doubt this is even a topology because the intersection of any two preimages from different $i$'s does not belong to $\tau$.

The only way I can relate $\tau$ with a topology is taking it as a subbasis for the Cartesian product $\prod_k X_k$.

Is there something I am missing here? Is $\tau$ indeed a topology as in the ProofWiki definition?

Best Answer

If we have a countable family of spaces $(X_i, \tau_i), i \in \Bbb N$ then the ProofWiki definition is a special case of the general product (for any number of spaces). There we consider all projections $\pi_i: X \to X_i$, where $X = \prod_{i \in \Bbb N} X_i$ (as a set), and define the product topology $\tau_p$ to be the smallest topology on $X$ such that all $\pi_i$ become continuous. Basic facts:

  • Sets of the form $\pi_i^{-1}[U_i]$, where $i \in \Bbb N$ and $U_i \in \tau_i$, together form a subbase for $\tau_p$.
  • It follows that the standard base (from that subbase) consists of all sets of the form $$\pi_{i_1}^{-1}[U_{i_1}] \cap \ldots \cap \pi_{i_m}^{-1}[U_{i_m}]$$ where $\{i_1, \ldots, i_m\} \subseteq \Bbb N$ is finite.
  • That set thus looks like (for the $\Bbb N$_case): $$X_1 \times \ldots \times U_{i_1} \times \ldots \times U_{i_2} \times \ldots \times U_{i_m} \times X_{{i_m}+1} \times \ldots$$ where all intermediate factors (not equal to some $U_{i_j}$) just are all full spaces $X_j$ for the right indices. Quite messy notation, but that's what the intersection means: finitely many coordinates are constrained (in that the points $(p_n)_n$ in the product must have $p_{i_j} \in U_{i_j}$ for a finite set of indices and corresponding open sets) and the others are free to choose in the correct factor space.

But when the index set is $\Bbb N$ and we have such a finite set of indices, we can just take that finite set to be $\{1, \ldots, k\}$ for some finite $k$ and we can just take it to be of the easier form

$$U_1 \times U_2 \times \ldots U_k \times X_{k+1} \times X_{k+2} \times \ldots$$ provided we allow for $U_i = X_i$ sometimes; then we can make all basic open subsets in this shape (cilinder sets) and then we're back at the OP's form of basic open sets. The OP's form is just a spacial case of the ProofWiki form and all ProofWiki's forms can be written as the OP's type. So they define exactly the same base and the same topology, even though they may look different at first sight....

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