# Topology/Product Spaces

Topology
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## 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 ${\displaystyle \Lambda }$  be an indexed set, and let ${\displaystyle X_{\lambda }}$  be a set for each ${\displaystyle \lambda \in \Lambda }$ . The Cartesian product of each ${\displaystyle X_{\lambda }}$  is

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

## ExampleEdit

Let ${\displaystyle \Lambda =\mathbb {N} }$  and ${\displaystyle X_{\lambda }=\mathbb {R} }$  for each ${\displaystyle n\in \mathbb {N} }$ . Then

${\displaystyle \prod _{\lambda \in \Lambda }X_{\lambda }=\mathbb {R} ^{\mathbb {N} }=\{x:\mathbb {N} \rightarrow \mathbb {R} \mid x(n)\in \mathbb {R} \,\forall \,n\in \mathbb {N} \}=\{(x_{1},x_{2},\ldots )\mid x_{n}\in \mathbb {R} \,\forall \,n\in \mathbb {N} \}}$ .

## Product TopologyEdit

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

### DefinitionEdit

Let ${\displaystyle X_{\lambda }}$  be a topological space. The product topology of ${\displaystyle \prod _{\lambda \in \Lambda }X_{\lambda }}$  is the topology with base elements of the form ${\displaystyle \prod _{\lambda \in \Lambda }U_{\lambda }}$ , where ${\displaystyle U_{\lambda }=X_{\lambda }}$  for all but a finite number of ${\displaystyle \lambda }$  and each ${\displaystyle U_{\lambda }}$  is open.

## ExamplesEdit

• Let ${\displaystyle \Lambda =\{1,2\}}$  and ${\displaystyle X_{\lambda }=\mathbb {R} }$  with the usual topology. Then the basic open sets of ${\displaystyle \mathbb {R} ^{2}}$  have the form ${\displaystyle (a,b)\times (c,d)}$ :

• Let ${\displaystyle \Lambda =\{1,2\}}$  and ${\displaystyle X_{\lambda }=R_{l}}$  (The Sorgenfrey topology). Then the basic open sets of ${\displaystyle \mathbb {R} ^{2}}$  are of the form ${\displaystyle [a,b)\times [a,b)}$ :

Topology
 ← Order Topology Product Spaces Quotient Spaces →