Product topology

Given a set [ilmath]X_{\alpha\in I} [/ilmath] of indexed topological spaces, we define the product topology, denoted [math]\prod_{\alpha\in I}X_\alpha[/math] (yes the Cartesian product) is the coarsest topology such that all the projection maps are continuous.

The projection maps are:

[math]p_\alpha:\prod_{\beta\in I}X_\beta\rightarrow X_\alpha[/math] which take the tuple [math](x_\alpha)_{\alpha\in I}\rightarrow x_{\beta}[/math]

This leads to the main property of the product topology, which can best be expressed as a diagram. Will add that later.

---

TODO:

---