Product topology

A base for a topology

1. Base of the product topology

   Let $X$ and $Y$ be topological spaces. Let $\mathcal{B}$ be the collection of all subsets of $X\times Y$ of the form $U\times V$, where $U$ is open in $X$ and $V$ is open in $Y$. Then $\mathcal{B}$ is a base for a topology on $X\times Y$.

2. Proof

   • For any $(x,y)\in X\times Y$, considering that $X\times Y\in \mathcal{B}$, we have that (B1) is satisfied.

   • Property $(B2)$ follows from the fact that in general:

     $$(A\times B)\cap (C\times D) = (A\cap C)\times (B\cap D). \qquad \Box$$

Product topology

1. Product topology

   The topology $\tau_{\mathcal{B}}$ is called the product topology on the Cartesian product $X\times Y$.

Product by a base

1. Product topology given by a base

   Let $X$ be space with base $\mathcal{B}$, and $Y$ be a space with base $\mathcal{C}$. Then

   $$\mathcal{D}=\{B\times C\mid B\in \mathcal{B}, C\in \mathcal{C}\}$$

   is a base for the product topology on $X\times Y$.

2. Proof

   • Let $W$ be an open set in the product topology, and $(x,y)\in W$.
   • By definition of base of the product topology, there are open sets $U$ in $X$ and $V$ in $Y$ with $(x,y)\in U\times V\subseteq W$.

   • Again by definition of base, there are $B\in \mathcal{B}$, $C\in \mathcal{C}$ with $x\in B\subseteq U$, $y\in C\subseteq V$. We have then $B\times C\in \mathcal{D}$, and

     $$(x,y)\in B\times C\subseteq W.\qquad \Box$$

Examples

1. Product topology on $\mathbb{R}^2$

   A base for the product of two standard topologies on $\mathbb{R}$ is the collection of all rectangles of the form $(a,b)\times (c,d)$. It follows that the product topology is the same as the metric topology on $\mathbb{R}$.

2. Product of Sierpiński spaces

   Describe the points and the open sets on the product $X\times X$, when $X$ is a Sierpiński space.

Projections

1. Projections

   The maps $\pi_{1}\colon X\times Y\to X$ and $\pi_{2}\colon X\times Y\to Y$ are called projections of $X\times Y$.

2. Inverse images of projections

   If $U$ is open in $X$, then $\pi_1^{-1}(U)=$

   $U\times Y$, which is open in $X\times Y$. Similarly, if $V$ is open in $Y$, the set $\pi_2^{-1}(V)=X\times V$, is open in $X\times Y$.

   We have

   $$\pi_1^{-1}(U)\cap \pi_2^{-1}(V) = U\times V$$

Subbase of the product

1. Subbase

   The collection $\mathcal{S}=\{\pi_1^{-1}(U)\}_{U\in\tau_{X}}\cup\{\pi_2^{-1}(V)\}_{V\in\tau_Y}$ is a subbase for the product topology on $X\times Y$.

Subspace and products

1. Subspace and product topology

   Let $A,B$ be subspaces of $X,Y$ respectively. Then the product topology on $A\times B$ is the same as the topology on $A\times B$ as subspace of the product space $X\times Y$.

2. Proof

   Exercise

Exercises

1. A map $f\colon X\to Y$ between topological space is said to be open if $f(U)$ is open for every $U\subseteq X$ which is an open set. Prove that the projection $\pi_{X}\colon X\times Y\to X$ is an open map.
2. Let $I$ denote the closed interval $[0,1]\subseteq \mathbb{R}$ as a subspace of the usual topology on $\mathbb{R}$. Compare the product topology on $I\times I$, the dictionary order topology on $I\times I$, and the product $I_{d}\times I$, where $I_{d}$ denotes discrete topology.

Links

• Product topology - Wikipedia, the free encyclopedia