Topology/Product Spaces

Before we begin

We quickly review the set-theoretic concept of Cartesian product here. This definition might be slightly more generalized than what you're used to.

Cartesian Product

Definition

Let $\Lambda$ be an indexed set, and let $X_\lambda$ be a set for each $\lambda \in \Lambda$. The Cartesian product of each $X_\lambda$ is

$\prod_{\lambda \in \Lambda}X_\lambda = \{x:\Lambda\rightarrow\bigcup_{i \in I} X_i | x(\lambda) \in X_\lambda\}$.

Example

Let $\Lambda = \mathbb{N}$ and $X_\lambda = \mathbb{R}$ for each $n \in \mathbb{N}$. Then

$\prod_{\lambda \in \Lambda} X_\lambda = \mathbb{R}^\mathbb{N} = \{x: \mathbb{N} \rightarrow \mathbb{R} \mid x(n) \in \mathbb{R}\, \forall\, n \in \mathbb{N}\} = \{(x_1, x_2, \ldots) \mid x_n \in \mathbb{R}\, \forall\, n \in \mathbb{N}\}$.

Product Topology

Using the Cartesian product, we can now define products of topological spaces.

Definition

Let $X_\lambda$ be a topological space. The product topology of $\prod_{\lambda \in \Lambda} X_\lambda$ is the topology with base elements of the form $\prod_{\lambda \in \Lambda} U_\lambda$, where $U_\lambda = X_\lambda$ for all but a finite number of $\lambda$ and each $U_\lambda$ is open.

Examples

• Let $\Lambda = \{1,2\}$ and $X_\lambda = \mathbb{R}$ with the usual topology. Then the basic open sets of $\mathbb{R}^2$ have the form $(a,b) \times (c,d)$:

• Let $\Lambda = \{1,2\}$ and $X_\lambda = R_l$ (The Sorgenfrey topology). Then the basic open sets of $\mathbb{R}^2$ are of the form $[a,b)\times [a,b)$: