Product Spaces

We want to take Cartesian product of a bunch of topological spaces, and give them a topology in some standard way.

Let $A$ be a non-empty set and for each $\alpha \in A$ , let $(X_\alpha, \tau_\alpha)$ be a topological space. Recall that the product space

$$\prod_{\alpha \in A} X_\alpha = \{ f : A \rightarrow \bigcup_{\alpha \in A} X_\alpha : f(\alpha) \in X_\alpha \forall \alpha \in A \}$$

Example: Suppose first that $A = \{ 1, 2, \ldots, n \}$ . Then,

$$\prod_{j=1}^n X_j = \{ f : \{ 1, \ldots, n \} \rightarrow \bigcup_{j=1}^n X_j : f(j) \in X_j \forall j \}$$

Write $x_j := f(j)$ . Then, we may view $f$ as the ordered $n$ -tuple $(x_1, \ldots, x_n)$ , where $x_j \in X_j$ . Hence,

When $(X_\alpha, \tau_\alpha)$ is a topological space, we give $$\prod_{\alpha \in A} X_\alpha$$ a topology from the following basis:

$$\mathcal{B}= \{ \prod_{\alpha \in A} G_\alpha : G_\alpha = X_\alpha$$    except for at most finitely many $\alpha$ , and $G_\alpha \in \tau_\alpha \forall \alpha \in A$ $$\}$$

The proof that this is a base is very similar to the proof of that for the topology of pointwise convergence.

Definition 6.24   Given the product $$\prod_{\alpha A} X_\alpha$$ and $\beta \in A$ , define the $\beta^{th}$ projection map

$$\pi_\beta: \prod_{\alpha \in A} X_\alpha \rightarrow X_\beta$$

by

$$\pi_\beta(f) = f(\beta)$$

Proposition 6.6   Each $\pi_\beta$ is continuous and moreover, the product topology is the smallest topology makin each $\pi_\beta$ continuous.