Product topology

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

The projection maps are:

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

This leads to the main property of the product topology, which can best be expressed as a diagram. As shown below:

10 x 5 = 50! this is a product

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