Product topology
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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:
