Topology/Product Spaces

We quickly review the set-theoretic concept of Cartesian product here. This definition might be slightly more generalized than what you're used to.

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

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

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

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

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

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.

• 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 ${\displaystyle [a,b)\times [a,b)}$: