Topology/Product Spaces

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Contents

Before we beginEdit

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 ProductEdit

DefinitionEdit

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


.

ExampleEdit

Let and for each . Then


.

Product TopologyEdit

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

DefinitionEdit

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.

ExamplesEdit

  • 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 :


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