# Topology/Product Spaces

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< Topology

## Before we begin[edit]

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[edit]

### Definition[edit]

Let be an indexed set, and let be a set for each . The **Cartesian product** of each is

## Example[edit]

Let and for each . Then

## Product Topology[edit]

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

### Definition[edit]

Let be a topological space. The **product topology** of is the topology with base elements of the form , where for all but a finite number of and each is open.

## Examples[edit]

- Let and with the usual topology. Then the basic open sets of have the form :

- Let and (The Sorgenfrey topology). Then the basic open sets of are of the form :