Product Topological Space

Last updated Nov 1, 2022

Let $I$ be a Nonempty Index Set and let ${X_{i}}{i \in I}$ be a collection of Topological Spaces. Then the Product Topological Space on $\prod\limits{i \in I} X_{i}$ is the topology generated by the set $$\mathcal{B} := {\pi_{i}^{-1}(U_{i}) : U_{i} \subset X_{i} \text{ is open}, i \in I}$$ where $\pi_{i}$ is the Projection Map $X \to X_{i}$ defined as $x \mapsto x_{i}$.

1. TODO - what is this saying? Should be stating $\mathcal{B}’$ is a Topological Basis, but looks like gibberish right now. Move to Product of Finitely Many Proper Open Sets form a Basis for the Product Topology. By The Topology Generated by a Set has a Basis of Finite Intersections, we know the Product Topological Space has basis$$\tau := {\prod\limits_{i \in I} U_{i} : U_{i} \subset X_{i} \text{ is open}, \text{ only finitely many }U_{i} \neq X_{i}, i \in I}$$ since $\prod\limits_{i \in I} U_{i} =\bigcap\limits_{i \in I, U_{i} \neq X_{i}} \pi_{i}^{-1}(U_{i})$ and only finitely many Open $U_{i} \neq X_{i}$.