Topology/Product Spaces

Before we begin

Template:TOC right 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 $\mathrm{\Lambda }$ be an indexed set, and let $X_{\lambda }$ be a set for each $\lambda \in \mathrm{\Lambda }$. The Cartesian product of each $X_{\lambda }$ is

$\prod _{\lambda \in \mathrm{\Lambda }}{X}_{\lambda }=\{x:\mathrm{\Lambda }\to \bigcup _{\lambda \in \mathrm{\Lambda }}{X}_{\lambda }|x(\lambda )\in X_{\lambda }\}$.

Example

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

$\prod _{\lambda \in \mathrm{\Lambda }}{X}_{\lambda }={ℝ}^{\mathbb{N}}=\{x:\mathbb{N}\to ℝ\mid x(n)\in ℝ\mathrm{\forall }n\in \mathbb{N}\}=\{(x_1,x_2,\dots )\mid x_n\in ℝ\mathrm{\forall }n\in \mathbb{N}\}$.

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 $\prod _{\lambda \in \mathrm{\Lambda }}{X}_{\lambda }$ is the topology with base elements of the form $\prod _{\lambda \in \mathrm{\Lambda }}{U}_{\lambda }$, where $U_{\lambda }=X_{\lambda }$ for all but a finite number of $\lambda$ and each $U_{\lambda }$ is open.

Examples

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

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