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.

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 $$[a,b)\times [a,b)$$: