# Topology/Product Spaces

 Topology ← Order Topology Product Spaces Quotient 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

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

## 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 $[a,b)\times [a,b)$ :

 Topology ← Order Topology Product Spaces Quotient Spaces →