The Product Topology

Topology · Axiom Academy

Combining topological spaces the right way Step 1: Building New Spaces from Old In topology, we often want to combine two topological spaces to create a new one. Given two topological spaces , we can form their Cartesian product as a set. But what topology should we put on this product set? Step 2: The Product Topology Definition Let and be topological spaces. The product topology on is the topology with basis . In other words, a basis element is any set of the form where and . This means that every open set in the product topology is a union of "rectangular" sets U × V, where U is open in X and V is open in Y. The product topology is the "right" definition because it makes the projection maps continuous . The projection maps are: To see why projections are continuous: If U is open in X, then is open in X × Y because it equals U × Y, which is a basis element (since Y is open in Y). Step 4: The Canonical Example — ℝ² The most important example is with the product topology, where both copies of ℝ have their standard topology. Basis for ℝ² with Product Topology: The basis consists of all sets of the form where and are open in ℝ. Since open sets in ℝ are unions of open intervals, the basis elements are open rectangles : products of open intervals. Step 5: Visualizing Open Rectangles Let's visualize how basis elements (open rectangles) look in ℝ². Each basis element is the Cartesian product of two open intervals.

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