2.5 Product Topologies
Recall that the Cartesian product \(X_1\times X_2\times ........\times X_n\) of sets \(X_1,X_2,.......,X_n\) is defined to be the set of all ordered n-tuples \((x_1,x_2,............,x_n)\), where \(x_i\in X_i\) for
\(i=1,2,.........,n\).
For example the sets \(\mathbb {R}^2\) and \(\mathbb {R}^3\) are the Cartesian products \(\mathbb {R}\times \mathbb {R}\) and \(\mathbb {R}\times \mathbb {R}\times \mathbb {R}\) respectively.
Definition 2.5.1. Let X and Y be topological spaces. A subset U of \(X\times Y\) is said to be open in \(X\times Y\) if
given any point (x,y) of U, there exists an open set V in X and an open set W in Y such that \(x\in V\)
and \(y\in W\) and \(V\times W\subseteq U\). The empty set is regarded as an open set in \(X\times Y\).
Theorem 2.5.2. Let X and Y be topological spaces. Then the collection of open sets in \(X\times Y\) is a
topology on \(X\times Y\).
Proof. The definition of open sets ensure that the empty set and the whole set \(X\times Y\) are open in \(X\times Y\).
Let \(E\) be the union of a collection of open sets in \(X\times Y\) and let \((x,y)\) be a point of \(E\). Then \((x,y)\in D\) for some open
set \(D\) in the collection. It follows from this that there exists an open set \(V\) in \(X\) and an open set \(W\) in \(Y\)
such that \(x\in V\), \(y\in W\) and \(V\times W\subseteq D\). But then \(V\times W\subseteq E\). It follows that \(E\) is open in \(X\times Y\).
Let \(U=U_1\cap U_2\cap ...........\cap U_m\), where \(U_1,U_2,..........,U_m\) are open sets in \(X\times Y\) and let \((x,y)\) be a point of \(U\). Then there exists open sets \(V_k\) in \(X\) and \(W_k\) in \(Y\)
for \(k=1,2,.............,m\) such that \(x\in V_k\), \(y\in W_k\) and \(V_k\times W_k\subseteq U_k\) for \(k=1,2,.........,m\). Let \(V=V_1\cap V_2\cap ................\cap V_m\) and \(W=W_1\cap W_2\cap ...............\cap W_m\). Then \(x\in V\) and \(y\in W\). Also, \(V\times W\subseteq V_k\times W_k\subseteq U_k\) for \(k=1,2,.............,m\). Thus \(V\times W\subseteq U\). It follows that \(U \)is open in
\(X\times Y\).
Let \(X\) and \(Y\) be topological spaces. The collection of open sets in \(X\times Y\) defined as described above is
referred to as the product topology on
\(X\times Y\). The definition of the product topology can be extended to Cartesian products of any finite
number of topological spaces.
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