Topology/Product Spaces

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Topology
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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 :

Opensetusual.jpg

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

Opensetsorg.jpg


Topology
 ← Order Topology Product Spaces Quotient Spaces →