Restricted Topological Products

In this section we describe a topological tool, which we need in order to define adeles (see Definition 20.3.1).

Definition 20.2.1 (Restricted Topological Products)   Let $X_\lambda$, for $\lambda\in\Lambda$, be a family of topological spaces, and for almost all $\lambda$ let $Y_{\lambda}\subset X_{\lambda}$ be an open subset of $X_{\lambda}$. Consider the space $X$ whose elements are sequences $\mathbf{x}= \{x_\lambda\}_{\lambda\in \Lambda}$, where $x_\lambda\in X_\lambda$ for every $\lambda$, and $x_\lambda\in Y_{\lambda}$ for almost all $\lambda$. We give $X$ a topology by taking as a basis of open sets the sets $\prod U_{\lambda}$, where $U_{\lambda}\subset X_{\lambda}$ is open for all $\lambda$, and $U_{\lambda} = Y_{\lambda}$ for almost all $\lambda$. We call $X$ with this topology the of the $X_{\lambda}$ with respect to the $Y_{\lambda}$.

Corollary 20.2.2   Let $S$ be a finite subset of $\Lambda$, and let $X_S$ be the set of $\mathbf{x}\in X$ with $x_\lambda\in Y_\lambda$ for all $\lambda\not\in S$, i.e.,

$$X_S = \prod_{\lambda \in S} X_{\lambda} \times \prod_{\lambda\not\in S} Y_{\lambda} \subset X.$$

Then $X_S$ is an open subset of $X$, and the topology induced on $X_S$ as a subset of $X$ is the same as the product topology.

The restricted topological product depends on the totality of the $Y_{\lambda}$, but not on the individual $Y_{\lambda}$:

Lemma 20.2.3   Let $Y_{\lambda}'\subset X_{\lambda}$ be open subsets, and suppose that $Y_{\lambda} = Y_{\lambda}'$ for almost all $\lambda$. Then the restricted topological product of the $X_\lambda$ with respect to the $Y_{\lambda}'$ is canonically isomorphic to the restricted topological product with respect to the $Y_{\lambda}$.

Lemma 20.2.4   Suppose that the $X_\lambda$ are locally compact and that the $Y_\lambda$ are compact. Then the restricted topological product $X$ of the $X_\lambda$ is locally compact.

Proof. For any finite subset $S$ of $\Lambda$, the open subset $X_S\subset X$ is locally compact, because by Lemma 20.2.2 it is a product of finitely many locally compact sets with an infinite product of compact sets. (Here we are using Tychonoff's theorem from topology, which asserts that an arbitrary product of compact topological spaces is compact (see Munkres's Topology, a first course, chapter 5).) Since $X=\cup_{S} X_S$, and the $X_S$ are open in $X$, the result follows. $\qedsymbol$

The following measure will be extremely important in deducing topological properties of the ideles, which will be used in proving finiteness of class groups. See, e.g., the proof of Lemma 20.4.1, which is a key input to the proof of strong approximation (Theorem 20.4.4).

Definition 20.2.5 (Product Measure)   For all $\lambda\in\Lambda$, suppose $\mu_\lambda$ is a measure on $X_\lambda$ with $\mu_\lambda(Y_\lambda) = 1$ when $Y_\lambda$ is defined. We define the $\mu$ on $X$ to be that for which a basis of measurable sets is

$$\prod_\lambda M_\lambda$$

where each $M_\lambda\subset X_\lambda$ has finite $\mu_\lambda$-measure and $M_\lambda=Y_\lambda$ for almost all $\lambda$, and where

$$\mu\left(\prod_\lambda M_\lambda\right) = \prod_\lambda \mu_\lambda(M_\lambda).$$