Section 15: The Product Topology on X×Y

The product topology on $X\times Y$ is the one generated by the basis consisting of all products $U\times V$ of open sets (or, equivalently, basis elements) $U\subset X$ and $V\subset Y$ .

If $U$ and $V$ are nonempty, then $U\times V$ is open in $X\times Y$ iff $U$ is open in $X$ and $V$ is open in $Y$ .
If $X$ and $Y$ are nonempty, and $\mathcal{T}$ and $\mathcal{T}'$ are topologies on $X$ , $\mathcal{U}$ and $\mathcal{U}'$ are topologies on $Y$ , then $\mathcal{T}\subset\mathcal{T}'$ and $\mathcal{U}\subset\mathcal{U}'$ (and at least one of the inclusions is proper) iff the topology of $X\times Y$ is (strictly) coarser than the topology of $X'\times Y'$ .

The projection of $X\times Y$ onto the first (second) factor, or coordinate, is a mapping given by $\pi_{1}(x,y)=x$ ( $\pi_{2}(x,y)=y$ ).

Another way to construct the product topology is to take as its subbasis the collection of all sets $\pi_{1}^{-1}(U)$ and $\pi_{2}^{-1}(V)$ where $U$ is open in $X$ and $V$ is open in $Y$ .

An open map $f:X\rightarrow Y$ is a function such that if $U$ is open in $X$ then $f(U)$ is open in $Y$ .

The projection is an example of an open map.