Topologies on the product

Products

1. Arbitrary Cartesian product as set

   Let $X_{\alpha}$ be a topological space for each $\alpha\in I$. The Cartesian product of the sets $X_{\alpha}$ is denoted as: [ \prod_{\alpha\in I}X_{\alpha} ] and consists of all maps $f\colon I\to \cup_{\alpha}X_{\alpha}$ such that $f(\alpha)\in X_{\alpha}$. If $f(\alpha)=x_{\alpha}$, we denote $f$ as $(x_{\alpha})_{\alpha}$.

2. Projections

   For each $\alpha_{0}\in I$, there is a projection map [ p_{\alpha_{0}}\colon\prod_{\alpha\in I}X_{\alpha}\to X_{\alpha_{0}}, ] given by $p_{\alpha_{0}}((x_{\alpha})_{\alpha})=x_{\alpha_{0}}$.

Box topology

1. Let $X_{\alpha}$ a topological space for each $\alpha\in I$, and $X$ the product $\prod X_{\alpha}$.

2. Box topology

   The collection of all subsets of $X$ of the form [ \prod_{\alpha\in I} U_{\alpha}, ] where $U_{\alpha}$ is open in $X_{\alpha}$ for all $\alpha\in I$, is a basis for a topology on $X$, called the box topology.

Product topology

1. Product topology

   The collection [ \mathcal{S}=\cup_{\alpha\in I}{p^{-1}{\alpha}(U{\alpha})\mid U_{\alpha}\text{ open in } X_{\alpha}} ] is a subbasis for a topology on $X$, called the product topology.

2. Remark

   The product topology has as basis the subsets of $X=\prod X_{\alpha}$ of the form [ \prod_{\alpha\in I} U_{\alpha}, ] where $U_{\alpha}$ is open in $X_{\alpha}$ for all $\alpha\in I$, and $U_{\alpha}=X_{\alpha}$ for all but finitely many values of $\alpha\in I$

Continuous functions into the product

1. Teorema

   Let $f\colon A\to\prod_{\alpha\in I}X_{\alpha}$ be given by: [ f(a)=(f_{\alpha}(a))_{\alpha\in I}, ] where $f_{\alpha}\colon A\to X_{\alpha}$ for each $\alpha$. Suppose that $\prod X_{\alpha}$ has the product topology. Then $f$ is continuous if and only if each $f_{\alpha}$ is continuous.

2. Remark

   The last theorem is not true if we use the box topology on the Cartesian product

Exercises

Exercises

1. Let $X_{\alpha}$ be a topological space for each $\alpha\in I$, and $A_{\alpha}\subseteq X_{\alpha}$.
   • Show that if $A_{\alpha}\subseteq X_{\alpha}$ is closed, then $\prod A_{\alpha}$ is closed in $\prod X_{\alpha}$.

   • Show that

     $$\overline{\prod A_{\alpha}}=\prod \overline{A_{\alpha}}.$$
   • Which of the last two are true if we use the box topology?
2. We say that the sequence $x_{n}$ of points in $X$ converges to $x\in X$ if for any neighborhood $U$ of $x$ there is $N$ such that $n\geq N$ implies $x_{n}\in U$. Show that a sequence $x_{n}$ in $\prod_{\alpha\in I} X_{\alpha}$ converges to $(x_{\alpha})$ if and only if $\pi_{\alpha}(x_{n})$ converges to $x_{\alpha}$ for each $\alpha\in I$.