Topology/Product Spaces

Before we begin

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

Definition

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

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

Example

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

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

Product Topology

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

Definition

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

Examples

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

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