Let \(X,Y\) be topological spaces, generated respectively by the bases \(\mc B_X,\mc B_Y\text{.}\) Then the product space is given by the set \(X\times Y=\setBuilder{\tuple{x,y}}{x\in X,y\in Y}\) with the topology generated by the basis \(\mc B=\setBuilder{U\times V}{U\in\mc B_X,V\in\mc B_Y}\text{.}\)
Proposition8.2.
The product spaces \(X\times Y\) and \(Y\times X\) are homeomorphic.
Proposition8.3.
Let \(p\in Y\text{.}\) The subspace \(X\times\{p\}=\setBuilder{\tuple{x,p}}{x\in X}\) of \(X\times Y\) is homeomorphic to \(X\text{.}\)
Theorem8.4.
The Euclidean space \(\mb R^{n+1}\) as defined in Definition 5.10 is homeomorphic to the product space \(\mb R^n\times\mb R\text{.}\)
Proposition8.5.
The product \(X\times Y\) is Hausdorff if and only if \(X,Y\) are each Hausdorff.
Theorem8.6.
The product \(X\times Y\) is regular if and only if \(X,Y\) are each regular.
Lemma8.7.
Let \(S=\mb R\) equipped with the Sorgenfrey topology. Then the product space \(S\times S\) contains two disjoint closed subsets \(H=\setBuilder{\tuple{x,-x}}{x\in\mb Q}\) and \(L=\setBuilder{\tuple{x,-x}}{x\in\mb R\setminus\mb Q}\) that cannot be separated by a pair of open sets.
Theorem8.8.
Let \(S=\mb R\) equipped with the Sorgenfrey topology. Then \(S\) is normal, but \(S\times S\) is not normal.
Proposition8.9.
The diagonal \(\Delta=\setBuilder{\tuple{x,x}}{x\in X}\) of \(X\times X\) is homeomorphic to \(X\text{.}\)
Definition8.10.
For a product space \(X\times Y\text{,}\) its projection maps \(\pi_X:X\times Y\to X\) and \(\pi_Y:X\times Y\to Y\) are defined by \(\pi_X(\tuple{x,y})=x\) and \(\pi_Y(\tuple{x,y})=y\text{.}\)
Checkpoint8.11.Properties of projection maps.
Verify the following properties of projection maps.
Every projection map is continuous.
The projection of an open set is always open.
The projection of a closed set is not always closed.
Theorem8.12.
The product \(X\times Y\) is metrizable if and only if \(X,Y\) are each metrizable.
Lemma8.13.
Let \(Y\) be compact. If \(\mc U\) is an open cover of \(X\times Y\text{,}\) then for each \(x\in X\) there exists a finite subcollection \(\mc F_x\subseteq\mc U\) and an open neighborhood \(U_x\) of \(x\) such that \(U_x\times Y\subseteq\bigcup\mc F_x\text{.}\)
Theorem8.14.
The product \(X\times Y\) is compact if and only if \(X,Y\) are each compact.
Theorem8.15.
The product \(X\times Y\) is connected if and only if \(X,Y\) are each connected.
