Topology/Product Spaces

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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  :

 

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

 


Topology
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